author | haftmann |
Sat, 05 Jul 2014 11:01:53 +0200 | |
changeset 57514 | bdc2c6b40bf2 |
parent 57418 | 6ab1c7cb0b8d |
child 58876 | 1888e3cb8048 |
permissions | -rw-r--r-- |
42067 | 1 |
(* Title: HOL/Probability/Information.thy |
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Author: Johannes Hölzl, TU München |
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Author: Armin Heller, TU München |
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*) |
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header {*Information theory*} |
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theory Information |
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imports |
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Independent_Family |
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"~~/src/HOL/Library/Convex" |
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begin |
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lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y" |
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by (subst log_le_cancel_iff) auto |
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lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y" |
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by (subst log_less_cancel_iff) auto |
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lemma setsum_cartesian_product': |
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"(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)" |
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unfolding setsum.cartesian_product by simp |
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lemma split_pairs: |
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"((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and |
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"(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto |
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subsection "Information theory" |
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locale information_space = prob_space + |
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fixes b :: real assumes b_gt_1: "1 < b" |
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context information_space |
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begin |
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text {* Introduce some simplification rules for logarithm of base @{term b}. *} |
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lemma log_neg_const: |
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assumes "x \<le> 0" |
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shows "log b x = log b 0" |
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proof - |
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{ fix u :: real |
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have "x \<le> 0" by fact |
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also have "0 < exp u" |
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using exp_gt_zero . |
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finally have "exp u \<noteq> x" |
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by auto } |
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then show "log b x = log b 0" |
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by (simp add: log_def ln_def) |
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qed |
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||
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lemma log_mult_eq: |
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"log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)" |
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using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"] |
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by (auto simp: zero_less_mult_iff mult_le_0_iff) |
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38656 | 56 |
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lemma log_inverse_eq: |
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"log b (inverse B) = (if 0 < B then - log b B else log b 0)" |
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using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp |
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lemma log_divide_eq: |
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"log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)" |
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unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse |
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by (auto simp: zero_less_mult_iff mult_le_0_iff) |
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38656 | 65 |
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lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq |
38656 | 67 |
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end |
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subsection "Kullback$-$Leibler divergence" |
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text {* The Kullback$-$Leibler divergence is also known as relative entropy or |
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Kullback$-$Leibler distance. *} |
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definition |
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47694 | 76 |
"entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N" |
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77 |
|
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definition |
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"KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)" |
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lemma measurable_entropy_density[measurable]: "entropy_density b M N \<in> borel_measurable M" |
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unfolding entropy_density_def by auto |
50003 | 83 |
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47694 | 84 |
lemma (in sigma_finite_measure) KL_density: |
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fixes f :: "'a \<Rightarrow> real" |
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assumes "1 < b" |
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assumes f[measurable]: "f \<in> borel_measurable M" and nn: "AE x in M. 0 \<le> f x" |
47694 | 88 |
shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)" |
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unfolding KL_divergence_def |
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proof (subst integral_real_density) |
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show [measurable]: "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M" |
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using f |
50003 | 93 |
by (auto simp: comp_def entropy_density_def) |
47694 | 94 |
have "density M (RN_deriv M (density M f)) = density M f" |
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using f nn by (intro density_RN_deriv_density) auto |
47694 | 96 |
then have eq: "AE x in M. RN_deriv M (density M f) x = f x" |
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using f nn by (intro density_unique) (auto simp: RN_deriv_nonneg) |
47694 | 98 |
show "(\<integral>x. f x * entropy_density b M (density M (\<lambda>x. ereal (f x))) x \<partial>M) = (\<integral>x. f x * log b (f x) \<partial>M)" |
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apply (intro integral_cong_AE) |
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apply measurable |
47694 | 101 |
using eq |
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apply eventually_elim |
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apply (auto simp: entropy_density_def) |
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done |
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qed fact+ |
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47694 | 107 |
lemma (in sigma_finite_measure) KL_density_density: |
108 |
fixes f g :: "'a \<Rightarrow> real" |
|
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assumes "1 < b" |
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assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
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assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
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assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
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shows "KL_divergence b (density M f) (density M g) = (\<integral>x. g x * log b (g x / f x) \<partial>M)" |
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proof - |
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interpret Mf: sigma_finite_measure "density M f" |
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using f by (subst sigma_finite_iff_density_finite) auto |
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have "KL_divergence b (density M f) (density M g) = |
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KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))" |
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using f g ac by (subst density_density_divide) simp_all |
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also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)" |
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121 |
using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density) |
47694 | 122 |
also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)" |
123 |
using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE) |
|
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finally show ?thesis . |
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qed |
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126 |
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47694 | 127 |
lemma (in information_space) KL_gt_0: |
128 |
fixes D :: "'a \<Rightarrow> real" |
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assumes "prob_space (density M D)" |
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assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
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assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
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assumes A: "density M D \<noteq> M" |
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shows "0 < KL_divergence b M (density M D)" |
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proof - |
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interpret N: prob_space "density M D" by fact |
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47694 | 137 |
obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A" |
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using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto |
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139 |
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let ?D_set = "{x\<in>space M. D x \<noteq> 0}" |
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have [simp, intro]: "?D_set \<in> sets M" |
|
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using D by auto |
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143 |
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have D_neg: "(\<integral>\<^sup>+ x. ereal (- D x) \<partial>M) = 0" |
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using D by (subst nn_integral_0_iff_AE) auto |
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146 |
|
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147 |
have "(\<integral>\<^sup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)" |
56996 | 148 |
using D by (simp add: emeasure_density cong: nn_integral_cong) |
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then have D_pos: "(\<integral>\<^sup>+ x. ereal (D x) \<partial>M) = 1" |
47694 | 150 |
using N.emeasure_space_1 by simp |
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151 |
|
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152 |
have "integrable M D" |
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153 |
using D D_pos D_neg unfolding real_integrable_def real_lebesgue_integral_def by simp_all |
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then have "integral\<^sup>L M D = 1" |
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using D D_pos D_neg by (simp add: real_lebesgue_integral_def) |
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156 |
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47694 | 157 |
have "0 \<le> 1 - measure M ?D_set" |
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158 |
using prob_le_1 by (auto simp: field_simps) |
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159 |
also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)" |
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160 |
using `integrable M D` `integral\<^sup>L M D = 1` |
47694 | 161 |
by (simp add: emeasure_eq_measure) |
162 |
also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)" |
|
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163 |
proof (rule integral_less_AE) |
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164 |
show "integrable M (\<lambda>x. D x - indicator ?D_set x)" |
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165 |
using `integrable M D` by auto |
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166 |
next |
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167 |
from integrable_mult_left(1)[OF int, of "ln b"] |
47694 | 168 |
show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" |
169 |
by (simp add: ac_simps) |
|
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170 |
next |
47694 | 171 |
show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0" |
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172 |
proof |
47694 | 173 |
assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0" |
174 |
then have disj: "AE x in M. D x = 1 \<or> D x = 0" |
|
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175 |
using D(1) by (auto intro!: AE_I[OF subset_refl] sets.sets_Collect) |
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|
176 |
|
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177 |
have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^sup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)" |
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|
178 |
using D(1) by auto |
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|
179 |
also have "\<dots> = (\<integral>\<^sup>+ x. ereal (D x) \<partial>M)" |
56996 | 180 |
using disj by (auto intro!: nn_integral_cong_AE simp: indicator_def one_ereal_def) |
47694 | 181 |
finally have "AE x in M. D x = 1" |
182 |
using D D_pos by (intro AE_I_eq_1) auto |
|
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|
183 |
then have "(\<integral>\<^sup>+x. indicator A x\<partial>M) = (\<integral>\<^sup>+x. ereal (D x) * indicator A x\<partial>M)" |
56996 | 184 |
by (intro nn_integral_cong_AE) (auto simp: one_ereal_def[symmetric]) |
47694 | 185 |
also have "\<dots> = density M D A" |
186 |
using `A \<in> sets M` D by (simp add: emeasure_density) |
|
187 |
finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp |
|
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|
188 |
qed |
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|
189 |
show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M" |
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immler
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50003
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|
190 |
using D(1) by (auto intro: sets.sets_Collect_conj) |
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|
191 |
|
47694 | 192 |
show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow> |
193 |
D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))" |
|
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|
194 |
using D(2) |
47694 | 195 |
proof (eventually_elim, safe) |
196 |
fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t" |
|
197 |
and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))" |
|
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|
198 |
|
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|
199 |
have "D t - 1 = D t - indicator ?D_set t" |
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|
200 |
using Dt by simp |
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|
201 |
also note eq |
47694 | 202 |
also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)" |
203 |
using b_gt_1 `D t \<noteq> 0` `0 \<le> D t` |
|
204 |
by (simp add: log_def ln_div less_le) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
205 |
finally have "ln (1 / D t) = 1 / D t - 1" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
206 |
using `D t \<noteq> 0` by (auto simp: field_simps) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
207 |
from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1` |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
208 |
show False by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
209 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
210 |
|
47694 | 211 |
show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
212 |
using D(2) AE_space |
|
213 |
proof eventually_elim |
|
214 |
fix t assume "t \<in> space M" "0 \<le> D t" |
|
215 |
show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
216 |
proof cases |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
217 |
assume asm: "D t \<noteq> 0" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
218 |
then have "0 < D t" using `0 \<le> D t` by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
219 |
then have "0 < 1 / D t" by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
220 |
have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
221 |
using asm `t \<in> space M` by (simp add: field_simps) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
222 |
also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
223 |
using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto |
47694 | 224 |
also have "\<dots> = D t * (ln b * log b (D t))" |
225 |
using `0 < D t` b_gt_1 |
|
226 |
by (simp_all add: log_def ln_div) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
227 |
finally show ?thesis by simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
228 |
qed simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
229 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
230 |
qed |
47694 | 231 |
also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)" |
232 |
by (simp add: ac_simps) |
|
233 |
also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
234 |
using int by simp |
47694 | 235 |
finally show ?thesis |
236 |
using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
237 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
238 |
|
47694 | 239 |
lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
240 |
proof - |
47694 | 241 |
have "AE x in M. 1 = RN_deriv M M x" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
242 |
proof (rule RN_deriv_unique) |
47694 | 243 |
show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto |
244 |
show "density M (\<lambda>x. 1) = M" |
|
245 |
apply (auto intro!: measure_eqI emeasure_density) |
|
246 |
apply (subst emeasure_density) |
|
247 |
apply auto |
|
248 |
done |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
249 |
qed |
47694 | 250 |
then have "AE x in M. log b (real (RN_deriv M M x)) = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
251 |
by (elim AE_mp) simp |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
252 |
from integral_cong_AE[OF _ _ this] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
253 |
have "integral\<^sup>L M (entropy_density b M M) = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
254 |
by (simp add: entropy_density_def comp_def) |
47694 | 255 |
then show "KL_divergence b M M = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
256 |
unfolding KL_divergence_def |
47694 | 257 |
by auto |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
258 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
259 |
|
47694 | 260 |
lemma (in information_space) KL_eq_0_iff_eq: |
261 |
fixes D :: "'a \<Rightarrow> real" |
|
262 |
assumes "prob_space (density M D)" |
|
263 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
264 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
265 |
shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M" |
|
266 |
using KL_same_eq_0[of b] KL_gt_0[OF assms] |
|
267 |
by (auto simp: less_le) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
268 |
|
47694 | 269 |
lemma (in information_space) KL_eq_0_iff_eq_ac: |
270 |
fixes D :: "'a \<Rightarrow> real" |
|
271 |
assumes "prob_space N" |
|
272 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
|
273 |
assumes int: "integrable N (entropy_density b M N)" |
|
274 |
shows "KL_divergence b M N = 0 \<longleftrightarrow> N = M" |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
275 |
proof - |
47694 | 276 |
interpret N: prob_space N by fact |
277 |
have "finite_measure N" by unfold_locales |
|
278 |
from real_RN_deriv[OF this ac] guess D . note D = this |
|
279 |
||
280 |
have "N = density M (RN_deriv M N)" |
|
281 |
using ac by (rule density_RN_deriv[symmetric]) |
|
282 |
also have "\<dots> = density M D" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
283 |
using D by (auto intro!: density_cong) |
47694 | 284 |
finally have N: "N = density M D" . |
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
285 |
|
47694 | 286 |
from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density |
287 |
have "integrable N (\<lambda>x. log b (D x))" |
|
288 |
by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int]) |
|
289 |
(auto simp: N entropy_density_def) |
|
290 |
with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
291 |
by (subst integrable_real_density[symmetric]) (auto simp: N[symmetric] comp_def) |
47694 | 292 |
with `prob_space N` D show ?thesis |
293 |
unfolding N |
|
294 |
by (intro KL_eq_0_iff_eq) auto |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
295 |
qed |
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
296 |
|
47694 | 297 |
lemma (in information_space) KL_nonneg: |
298 |
assumes "prob_space (density M D)" |
|
299 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
300 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
301 |
shows "0 \<le> KL_divergence b M (density M D)" |
|
302 |
using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0) |
|
40859 | 303 |
|
47694 | 304 |
lemma (in sigma_finite_measure) KL_density_density_nonneg: |
305 |
fixes f g :: "'a \<Rightarrow> real" |
|
306 |
assumes "1 < b" |
|
307 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)" |
|
308 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)" |
|
309 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
310 |
assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))" |
|
311 |
shows "0 \<le> KL_divergence b (density M f) (density M g)" |
|
312 |
proof - |
|
313 |
interpret Mf: prob_space "density M f" by fact |
|
314 |
interpret Mf: information_space "density M f" b by default fact |
|
315 |
have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _") |
|
316 |
using f g ac by (subst density_density_divide) simp_all |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
317 |
|
47694 | 318 |
have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))" |
319 |
proof (rule Mf.KL_nonneg) |
|
320 |
show "prob_space ?DD" unfolding eq by fact |
|
321 |
from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)" |
|
322 |
by auto |
|
323 |
show "AE x in density M f. 0 \<le> g x / f x" |
|
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
324 |
using f g by (auto simp: AE_density) |
47694 | 325 |
show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))" |
326 |
using `1 < b` f g ac |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
327 |
by (subst integrable_density) |
47694 | 328 |
(auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If) |
329 |
qed |
|
330 |
also have "\<dots> = KL_divergence b (density M f) (density M g)" |
|
331 |
using f g ac by (subst density_density_divide) simp_all |
|
332 |
finally show ?thesis . |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
333 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
334 |
|
49803 | 335 |
subsection {* Finite Entropy *} |
336 |
||
337 |
definition (in information_space) |
|
338 |
"finite_entropy S X f \<longleftrightarrow> distributed M S X f \<and> integrable S (\<lambda>x. f x * log b (f x))" |
|
339 |
||
340 |
lemma (in information_space) finite_entropy_simple_function: |
|
341 |
assumes X: "simple_function M X" |
|
342 |
shows "finite_entropy (count_space (X`space M)) X (\<lambda>a. measure M {x \<in> space M. X x = a})" |
|
343 |
unfolding finite_entropy_def |
|
344 |
proof |
|
345 |
have [simp]: "finite (X ` space M)" |
|
346 |
using X by (auto simp: simple_function_def) |
|
347 |
then show "integrable (count_space (X ` space M)) |
|
348 |
(\<lambda>x. prob {xa \<in> space M. X xa = x} * log b (prob {xa \<in> space M. X xa = x}))" |
|
349 |
by (rule integrable_count_space) |
|
350 |
have d: "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then prob {xa \<in> space M. X xa = x} else 0))" |
|
351 |
by (rule distributed_simple_function_superset[OF X]) (auto intro!: arg_cong[where f=prob]) |
|
352 |
show "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (prob {xa \<in> space M. X xa = x}))" |
|
353 |
by (rule distributed_cong_density[THEN iffD1, OF _ _ _ d]) auto |
|
354 |
qed |
|
355 |
||
356 |
lemma distributed_transform_AE: |
|
357 |
assumes T: "T \<in> measurable P Q" "absolutely_continuous Q (distr P Q T)" |
|
358 |
assumes g: "distributed M Q Y g" |
|
359 |
shows "AE x in P. 0 \<le> g (T x)" |
|
360 |
using g |
|
361 |
apply (subst AE_distr_iff[symmetric, OF T(1)]) |
|
50003 | 362 |
apply simp |
49803 | 363 |
apply (rule absolutely_continuous_AE[OF _ T(2)]) |
364 |
apply simp |
|
365 |
apply (simp add: distributed_AE) |
|
366 |
done |
|
367 |
||
368 |
lemma ac_fst: |
|
369 |
assumes "sigma_finite_measure T" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
370 |
shows "absolutely_continuous S (distr (S \<Otimes>\<^sub>M T) S fst)" |
49803 | 371 |
proof - |
372 |
interpret sigma_finite_measure T by fact |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
373 |
{ fix A assume A: "A \<in> sets S" "emeasure S A = 0" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
374 |
then have "fst -` A \<inter> space (S \<Otimes>\<^sub>M T) = A \<times> space T" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50003
diff
changeset
|
375 |
by (auto simp: space_pair_measure dest!: sets.sets_into_space) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
376 |
with A have "emeasure (S \<Otimes>\<^sub>M T) (fst -` A \<inter> space (S \<Otimes>\<^sub>M T)) = 0" |
49803 | 377 |
by (simp add: emeasure_pair_measure_Times) } |
378 |
then show ?thesis |
|
379 |
unfolding absolutely_continuous_def |
|
380 |
apply (auto simp: null_sets_distr_iff) |
|
381 |
apply (auto simp: null_sets_def intro!: measurable_sets) |
|
382 |
done |
|
383 |
qed |
|
384 |
||
385 |
lemma ac_snd: |
|
386 |
assumes "sigma_finite_measure T" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
387 |
shows "absolutely_continuous T (distr (S \<Otimes>\<^sub>M T) T snd)" |
49803 | 388 |
proof - |
389 |
interpret sigma_finite_measure T by fact |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
390 |
{ fix A assume A: "A \<in> sets T" "emeasure T A = 0" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
391 |
then have "snd -` A \<inter> space (S \<Otimes>\<^sub>M T) = space S \<times> A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50003
diff
changeset
|
392 |
by (auto simp: space_pair_measure dest!: sets.sets_into_space) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
393 |
with A have "emeasure (S \<Otimes>\<^sub>M T) (snd -` A \<inter> space (S \<Otimes>\<^sub>M T)) = 0" |
49803 | 394 |
by (simp add: emeasure_pair_measure_Times) } |
395 |
then show ?thesis |
|
396 |
unfolding absolutely_continuous_def |
|
397 |
apply (auto simp: null_sets_distr_iff) |
|
398 |
apply (auto simp: null_sets_def intro!: measurable_sets) |
|
399 |
done |
|
400 |
qed |
|
401 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
402 |
lemma integrable_cong_AE_imp: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
403 |
"integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
404 |
using integrable_cong_AE[of f M g] by (auto simp: eq_commute) |
49803 | 405 |
|
406 |
lemma (in information_space) finite_entropy_integrable: |
|
407 |
"finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))" |
|
408 |
unfolding finite_entropy_def by auto |
|
409 |
||
410 |
lemma (in information_space) finite_entropy_distributed: |
|
411 |
"finite_entropy S X Px \<Longrightarrow> distributed M S X Px" |
|
412 |
unfolding finite_entropy_def by auto |
|
413 |
||
414 |
lemma (in information_space) finite_entropy_integrable_transform: |
|
415 |
assumes Fx: "finite_entropy S X Px" |
|
416 |
assumes Fy: "distributed M T Y Py" |
|
417 |
and "X = (\<lambda>x. f (Y x))" |
|
418 |
and "f \<in> measurable T S" |
|
419 |
shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))" |
|
420 |
using assms unfolding finite_entropy_def |
|
421 |
using distributed_transform_integrable[of M T Y Py S X Px f "\<lambda>x. log b (Px x)"] |
|
50003 | 422 |
by auto |
49803 | 423 |
|
39097 | 424 |
subsection {* Mutual Information *} |
425 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
426 |
definition (in prob_space) |
38656 | 427 |
"mutual_information b S T X Y = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
428 |
KL_divergence b (distr M S X \<Otimes>\<^sub>M distr M T Y) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
429 |
|
47694 | 430 |
lemma (in information_space) mutual_information_indep_vars: |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
431 |
fixes S T X Y |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
432 |
defines "P \<equiv> distr M S X \<Otimes>\<^sub>M distr M T Y" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
433 |
defines "Q \<equiv> distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
434 |
shows "indep_var S X T Y \<longleftrightarrow> |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
435 |
(random_variable S X \<and> random_variable T Y \<and> |
47694 | 436 |
absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and> |
437 |
mutual_information b S T X Y = 0)" |
|
438 |
unfolding indep_var_distribution_eq |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
439 |
proof safe |
50003 | 440 |
assume rv[measurable]: "random_variable S X" "random_variable T Y" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
441 |
|
47694 | 442 |
interpret X: prob_space "distr M S X" |
443 |
by (rule prob_space_distr) fact |
|
444 |
interpret Y: prob_space "distr M T Y" |
|
445 |
by (rule prob_space_distr) fact |
|
446 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default |
|
447 |
interpret P: information_space P b unfolding P_def by default (rule b_gt_1) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
448 |
|
47694 | 449 |
interpret Q: prob_space Q unfolding Q_def |
50003 | 450 |
by (rule prob_space_distr) simp |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
451 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
452 |
{ assume "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" |
47694 | 453 |
then have [simp]: "Q = P" unfolding Q_def P_def by simp |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
454 |
|
47694 | 455 |
show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def) |
456 |
then have ed: "entropy_density b P Q \<in> borel_measurable P" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
457 |
by simp |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
458 |
|
47694 | 459 |
have "AE x in P. 1 = RN_deriv P Q x" |
460 |
proof (rule P.RN_deriv_unique) |
|
461 |
show "density P (\<lambda>x. 1) = Q" |
|
462 |
unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density) |
|
463 |
qed auto |
|
464 |
then have ae_0: "AE x in P. entropy_density b P Q x = 0" |
|
465 |
by eventually_elim (auto simp: entropy_density_def) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
466 |
then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0::real)" |
47694 | 467 |
using ed unfolding `Q = P` by (intro integrable_cong_AE) auto |
468 |
then show "integrable Q (entropy_density b P Q)" by simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
469 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
470 |
from ae_0 have "mutual_information b S T X Y = (\<integral>x. 0 \<partial>P)" |
47694 | 471 |
unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P` |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
472 |
by (intro integral_cong_AE) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
473 |
then show "mutual_information b S T X Y = 0" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
474 |
by simp } |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
475 |
|
47694 | 476 |
{ assume ac: "absolutely_continuous P Q" |
477 |
assume int: "integrable Q (entropy_density b P Q)" |
|
478 |
assume I_eq_0: "mutual_information b S T X Y = 0" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
479 |
|
47694 | 480 |
have eq: "Q = P" |
481 |
proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1]) |
|
482 |
show "prob_space Q" by unfold_locales |
|
483 |
show "absolutely_continuous P Q" by fact |
|
484 |
show "integrable Q (entropy_density b P Q)" by fact |
|
485 |
show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure) |
|
486 |
show "KL_divergence b P Q = 0" |
|
487 |
using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def) |
|
488 |
qed |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
489 |
then show "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" |
47694 | 490 |
unfolding P_def Q_def .. } |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
491 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
492 |
|
40859 | 493 |
abbreviation (in information_space) |
494 |
mutual_information_Pow ("\<I>'(_ ; _')") where |
|
47694 | 495 |
"\<I>(X ; Y) \<equiv> mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
496 |
|
47694 | 497 |
lemma (in information_space) |
498 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
49803 | 499 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
500 |
assumes Fx: "finite_entropy S X Px" and Fy: "finite_entropy T Y Py" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
501 |
assumes Fxy: "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
49803 | 502 |
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
503 |
shows mutual_information_distr': "mutual_information b S T X Y = integral\<^sup>L (S \<Otimes>\<^sub>M T) f" (is "?M = ?R") |
49803 | 504 |
and mutual_information_nonneg': "0 \<le> mutual_information b S T X Y" |
505 |
proof - |
|
506 |
have Px: "distributed M S X Px" |
|
507 |
using Fx by (auto simp: finite_entropy_def) |
|
508 |
have Py: "distributed M T Y Py" |
|
509 |
using Fy by (auto simp: finite_entropy_def) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
510 |
have Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
49803 | 511 |
using Fxy by (auto simp: finite_entropy_def) |
512 |
||
513 |
have X: "random_variable S X" |
|
50003 | 514 |
using Px by auto |
49803 | 515 |
have Y: "random_variable T Y" |
50003 | 516 |
using Py by auto |
49803 | 517 |
interpret S: sigma_finite_measure S by fact |
518 |
interpret T: sigma_finite_measure T by fact |
|
519 |
interpret ST: pair_sigma_finite S T .. |
|
520 |
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr) |
|
521 |
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr) |
|
522 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
523 |
let ?P = "S \<Otimes>\<^sub>M T" |
49803 | 524 |
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))" |
525 |
||
526 |
{ fix A assume "A \<in> sets S" |
|
527 |
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)" |
|
528 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
529 |
intro!: arg_cong[where f="emeasure M"]) } |
|
530 |
note marginal_eq1 = this |
|
531 |
{ fix A assume "A \<in> sets T" |
|
532 |
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)" |
|
533 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
534 |
intro!: arg_cong[where f="emeasure M"]) } |
|
535 |
note marginal_eq2 = this |
|
536 |
||
537 |
have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))" |
|
538 |
by auto |
|
539 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
540 |
have distr_eq: "distr M S X \<Otimes>\<^sub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))" |
49803 | 541 |
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq |
542 |
proof (subst pair_measure_density) |
|
543 |
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T" |
|
544 |
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)" |
|
545 |
using Px Py by (auto simp: distributed_def) |
|
546 |
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
547 |
qed (fact | simp)+ |
|
548 |
||
549 |
have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))" |
|
550 |
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] .. |
|
551 |
||
552 |
from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P" |
|
553 |
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'') |
|
554 |
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)" |
|
555 |
proof (rule ST.AE_pair_measure) |
|
556 |
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P" |
|
557 |
using f by auto |
|
558 |
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))" |
|
559 |
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE) |
|
560 |
qed |
|
561 |
||
562 |
have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)" |
|
563 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
564 |
moreover |
|
565 |
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)" |
|
566 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
567 |
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
568 |
by eventually_elim auto |
|
569 |
||
570 |
show "?M = ?R" |
|
571 |
unfolding M f_def |
|
572 |
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac |
|
573 |
by (rule ST.KL_density_density) |
|
574 |
||
575 |
have X: "X = fst \<circ> (\<lambda>x. (X x, Y x))" and Y: "Y = snd \<circ> (\<lambda>x. (X x, Y x))" |
|
576 |
by auto |
|
577 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
578 |
have "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)))" |
49803 | 579 |
using finite_entropy_integrable[OF Fxy] |
580 |
using finite_entropy_integrable_transform[OF Fx Pxy, of fst] |
|
581 |
using finite_entropy_integrable_transform[OF Fy Pxy, of snd] |
|
582 |
by simp |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
583 |
moreover have "f \<in> borel_measurable (S \<Otimes>\<^sub>M T)" |
49803 | 584 |
unfolding f_def using Px Py Pxy |
585 |
by (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'' |
|
586 |
intro!: borel_measurable_times borel_measurable_log borel_measurable_divide) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
587 |
ultimately have int: "integrable (S \<Otimes>\<^sub>M T) f" |
49803 | 588 |
apply (rule integrable_cong_AE_imp) |
589 |
using |
|
590 |
distributed_transform_AE[OF measurable_fst ac_fst, of T, OF T Px] |
|
591 |
distributed_transform_AE[OF measurable_snd ac_snd, of _ _ _ _ S, OF T Py] |
|
592 |
subdensity_real[OF measurable_fst Pxy Px X] |
|
593 |
subdensity_real[OF measurable_snd Pxy Py Y] |
|
594 |
distributed_real_AE[OF Pxy] |
|
595 |
by eventually_elim |
|
56536 | 596 |
(auto simp: f_def log_divide_eq log_mult_eq field_simps zero_less_mult_iff) |
49803 | 597 |
|
598 |
show "0 \<le> ?M" unfolding M |
|
599 |
proof (rule ST.KL_density_density_nonneg |
|
600 |
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]]) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
601 |
show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x))) " |
49803 | 602 |
unfolding distributed_distr_eq_density[OF Pxy, symmetric] |
603 |
using distributed_measurable[OF Pxy] by (rule prob_space_distr) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
604 |
show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))" |
49803 | 605 |
unfolding distr_eq[symmetric] by unfold_locales |
606 |
qed |
|
607 |
qed |
|
608 |
||
609 |
||
610 |
lemma (in information_space) |
|
611 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
47694 | 612 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
613 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
614 |
assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
47694 | 615 |
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
616 |
shows mutual_information_distr: "mutual_information b S T X Y = integral\<^sup>L (S \<Otimes>\<^sub>M T) f" (is "?M = ?R") |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
617 |
and mutual_information_nonneg: "integrable (S \<Otimes>\<^sub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y" |
40859 | 618 |
proof - |
47694 | 619 |
have X: "random_variable S X" |
620 |
using Px by (auto simp: distributed_def) |
|
621 |
have Y: "random_variable T Y" |
|
622 |
using Py by (auto simp: distributed_def) |
|
623 |
interpret S: sigma_finite_measure S by fact |
|
624 |
interpret T: sigma_finite_measure T by fact |
|
625 |
interpret ST: pair_sigma_finite S T .. |
|
626 |
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr) |
|
627 |
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr) |
|
628 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
629 |
let ?P = "S \<Otimes>\<^sub>M T" |
47694 | 630 |
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))" |
631 |
||
632 |
{ fix A assume "A \<in> sets S" |
|
633 |
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)" |
|
634 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
635 |
intro!: arg_cong[where f="emeasure M"]) } |
|
636 |
note marginal_eq1 = this |
|
637 |
{ fix A assume "A \<in> sets T" |
|
638 |
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)" |
|
639 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
640 |
intro!: arg_cong[where f="emeasure M"]) } |
|
641 |
note marginal_eq2 = this |
|
642 |
||
643 |
have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))" |
|
644 |
by auto |
|
645 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
646 |
have distr_eq: "distr M S X \<Otimes>\<^sub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))" |
47694 | 647 |
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq |
648 |
proof (subst pair_measure_density) |
|
649 |
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T" |
|
650 |
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)" |
|
651 |
using Px Py by (auto simp: distributed_def) |
|
652 |
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
653 |
qed (fact | simp)+ |
|
654 |
||
655 |
have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))" |
|
656 |
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] .. |
|
657 |
||
658 |
from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P" |
|
659 |
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'') |
|
660 |
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)" |
|
661 |
proof (rule ST.AE_pair_measure) |
|
662 |
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P" |
|
663 |
using f by auto |
|
664 |
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))" |
|
665 |
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE) |
|
666 |
qed |
|
667 |
||
668 |
have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)" |
|
669 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
670 |
moreover |
|
671 |
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)" |
|
672 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
673 |
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
674 |
by eventually_elim auto |
|
675 |
||
676 |
show "?M = ?R" |
|
677 |
unfolding M f_def |
|
678 |
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac |
|
679 |
by (rule ST.KL_density_density) |
|
680 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
681 |
assume int: "integrable (S \<Otimes>\<^sub>M T) f" |
47694 | 682 |
show "0 \<le> ?M" unfolding M |
683 |
proof (rule ST.KL_density_density_nonneg |
|
684 |
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]]) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
685 |
show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x))) " |
47694 | 686 |
unfolding distributed_distr_eq_density[OF Pxy, symmetric] |
687 |
using distributed_measurable[OF Pxy] by (rule prob_space_distr) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
688 |
show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))" |
47694 | 689 |
unfolding distr_eq[symmetric] by unfold_locales |
40859 | 690 |
qed |
691 |
qed |
|
692 |
||
693 |
lemma (in information_space) |
|
47694 | 694 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
695 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
696 |
assumes Px[measurable]: "distributed M S X Px" and Py[measurable]: "distributed M T Y Py" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
697 |
assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
47694 | 698 |
assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y" |
699 |
shows mutual_information_eq_0: "mutual_information b S T X Y = 0" |
|
36624 | 700 |
proof - |
47694 | 701 |
interpret S: sigma_finite_measure S by fact |
702 |
interpret T: sigma_finite_measure T by fact |
|
703 |
interpret ST: pair_sigma_finite S T .. |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
704 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
705 |
have "AE x in S \<Otimes>\<^sub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 706 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
707 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
708 |
have "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 709 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
710 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
711 |
have "AE x in S \<Otimes>\<^sub>M T. Pxy x = Px (fst x) * Py (snd x)" |
47694 | 712 |
using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy] |
713 |
by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'') |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
714 |
ultimately have "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0" |
47694 | 715 |
by eventually_elim simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
716 |
then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^sub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^sub>M T))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
717 |
by (intro integral_cong_AE) auto |
47694 | 718 |
then show ?thesis |
719 |
by (subst mutual_information_distr[OF assms(1-5)]) simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
720 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
721 |
|
47694 | 722 |
lemma (in information_space) mutual_information_simple_distributed: |
723 |
assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py" |
|
724 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
725 |
shows "\<I>(X ; Y) = (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x))`space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))" |
|
726 |
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
|
727 |
note fin = simple_distributed_joint_finite[OF XY, simp] |
|
728 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
729 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
730 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
731 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
732 |
let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
733 |
let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))" |
|
734 |
have "\<And>x. ?f x = (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) else 0)" |
|
735 |
by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
736 |
with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M))) = |
47694 | 737 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))" |
57418 | 738 |
by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum.If_cases split_beta' |
739 |
intro!: setsum.cong) |
|
47694 | 740 |
qed |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
741 |
|
47694 | 742 |
lemma (in information_space) |
743 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
744 |
assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py" |
|
745 |
assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
746 |
assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)" |
|
747 |
shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0" |
|
748 |
proof (subst mutual_information_simple_distributed[OF Px Py Pxy]) |
|
749 |
have "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = |
|
750 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)" |
|
57418 | 751 |
by (intro setsum.cong) (auto simp: ae) |
47694 | 752 |
then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. |
753 |
Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp |
|
754 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
755 |
|
39097 | 756 |
subsection {* Entropy *} |
757 |
||
47694 | 758 |
definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where |
759 |
"entropy b S X = - KL_divergence b S (distr M S X)" |
|
760 |
||
40859 | 761 |
abbreviation (in information_space) |
762 |
entropy_Pow ("\<H>'(_')") where |
|
47694 | 763 |
"\<H>(X) \<equiv> entropy b (count_space (X`space M)) X" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41833
diff
changeset
|
764 |
|
49791 | 765 |
lemma (in prob_space) distributed_RN_deriv: |
766 |
assumes X: "distributed M S X Px" |
|
767 |
shows "AE x in S. RN_deriv S (density S Px) x = Px x" |
|
768 |
proof - |
|
769 |
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
770 |
interpret X: prob_space "distr M S X" |
|
771 |
using D(1) by (rule prob_space_distr) |
|
772 |
||
773 |
have sf: "sigma_finite_measure (distr M S X)" by default |
|
774 |
show ?thesis |
|
775 |
using D |
|
776 |
apply (subst eq_commute) |
|
777 |
apply (intro RN_deriv_unique_sigma_finite) |
|
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
778 |
apply (auto simp: distributed_distr_eq_density[symmetric, OF X] sf measure_nonneg) |
49791 | 779 |
done |
780 |
qed |
|
781 |
||
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
782 |
lemma (in information_space) |
47694 | 783 |
fixes X :: "'a \<Rightarrow> 'b" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
784 |
assumes X[measurable]: "distributed M MX X f" |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
785 |
shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq) |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
786 |
proof - |
49785 | 787 |
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X] |
49791 | 788 |
note ae = distributed_RN_deriv[OF X] |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
789 |
|
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
790 |
have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) = |
49785 | 791 |
log b (f x)" |
792 |
unfolding distributed_distr_eq_density[OF X] |
|
793 |
apply (subst AE_density) |
|
794 |
using D apply simp |
|
795 |
using ae apply eventually_elim |
|
796 |
apply auto |
|
797 |
done |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
798 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
799 |
have int_eq: "(\<integral> x. f x * log b (f x) \<partial>MX) = (\<integral> x. log b (f x) \<partial>distr M MX X)" |
49785 | 800 |
unfolding distributed_distr_eq_density[OF X] |
801 |
using D |
|
802 |
by (subst integral_density) |
|
803 |
(auto simp: borel_measurable_ereal_iff) |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
804 |
|
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
805 |
show ?eq |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
806 |
unfolding entropy_def KL_divergence_def entropy_density_def comp_def int_eq neg_equal_iff_equal |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
807 |
using ae_eq by (intro integral_cong_AE) auto |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
808 |
qed |
49785 | 809 |
|
49786 | 810 |
lemma (in prob_space) distributed_imp_emeasure_nonzero: |
811 |
assumes X: "distributed M MX X Px" |
|
812 |
shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0" |
|
813 |
proof |
|
814 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
815 |
interpret X: prob_space "distr M MX X" |
|
816 |
using distributed_measurable[OF X] by (rule prob_space_distr) |
|
817 |
||
818 |
assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0" |
|
819 |
with Px have "AE x in MX. Px x = 0" |
|
820 |
by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ereal_iff) |
|
821 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
822 |
from X.emeasure_space_1 have "(\<integral>\<^sup>+x. Px x \<partial>MX) = 1" |
49786 | 823 |
unfolding distributed_distr_eq_density[OF X] using Px |
824 |
by (subst (asm) emeasure_density) |
|
56996 | 825 |
(auto simp: borel_measurable_ereal_iff intro!: integral_cong cong: nn_integral_cong) |
49786 | 826 |
ultimately show False |
56996 | 827 |
by (simp add: nn_integral_cong_AE) |
49786 | 828 |
qed |
829 |
||
830 |
lemma (in information_space) entropy_le: |
|
831 |
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
832 |
assumes X[measurable]: "distributed M MX X Px" |
49786 | 833 |
and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>" |
834 |
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
835 |
shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})" |
|
836 |
proof - |
|
837 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
838 |
interpret X: prob_space "distr M MX X" |
|
839 |
using distributed_measurable[OF X] by (rule prob_space_distr) |
|
840 |
||
841 |
have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) = |
|
842 |
- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)" |
|
843 |
using Px fin |
|
57166
5cfcc616d485
use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents:
56996
diff
changeset
|
844 |
by (auto simp: measure_def borel_measurable_ereal_iff) |
49786 | 845 |
also have "- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX) = - log b (\<integral> x. 1 / Px x \<partial>distr M MX X)" |
846 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
847 |
apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus]) |
|
57166
5cfcc616d485
use 0 as integral-value for non-integrable functions, simplify a couple of rewrite rules
hoelzl
parents:
56996
diff
changeset
|
848 |
by (subst integral_density) (auto simp: borel_measurable_ereal_iff simp del: integral_indicator intro!: integral_cong) |
49786 | 849 |
also have "\<dots> \<le> (\<integral> x. - log b (1 / Px x) \<partial>distr M MX X)" |
850 |
proof (rule X.jensens_inequality[of "\<lambda>x. 1 / Px x" "{0<..}" 0 1 "\<lambda>x. - log b x"]) |
|
851 |
show "AE x in distr M MX X. 1 / Px x \<in> {0<..}" |
|
852 |
unfolding distributed_distr_eq_density[OF X] |
|
853 |
using Px by (auto simp: AE_density) |
|
854 |
have [simp]: "\<And>x. x \<in> space MX \<Longrightarrow> ereal (if Px x = 0 then 0 else 1) = indicator {x \<in> space MX. Px x \<noteq> 0} x" |
|
855 |
by (auto simp: one_ereal_def) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
856 |
have "(\<integral>\<^sup>+ x. max 0 (ereal (- (if Px x = 0 then 0 else 1))) \<partial>MX) = (\<integral>\<^sup>+ x. 0 \<partial>MX)" |
56996 | 857 |
by (intro nn_integral_cong) (auto split: split_max) |
49786 | 858 |
then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)" |
859 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
56996 | 860 |
by (auto simp: nn_integral_density real_integrable_def borel_measurable_ereal_iff fin nn_integral_max_0 |
861 |
cong: nn_integral_cong) |
|
49786 | 862 |
have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) = |
863 |
integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
864 |
using Px |
|
865 |
by (intro integrable_cong_AE) |
|
866 |
(auto simp: borel_measurable_ereal_iff log_divide_eq |
|
867 |
intro!: measurable_If) |
|
868 |
then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))" |
|
869 |
unfolding distributed_distr_eq_density[OF X] |
|
870 |
using Px int |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
871 |
by (subst integrable_real_density) (auto simp: borel_measurable_ereal_iff) |
49786 | 872 |
qed (auto simp: minus_log_convex[OF b_gt_1]) |
873 |
also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)" |
|
874 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
875 |
by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq) |
|
876 |
also have "\<dots> = - entropy b MX X" |
|
877 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
878 |
by (subst entropy_distr[OF X]) (auto simp: borel_measurable_ereal_iff integral_density) |
|
879 |
finally show ?thesis |
|
880 |
by simp |
|
881 |
qed |
|
882 |
||
883 |
lemma (in information_space) entropy_le_space: |
|
884 |
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure" |
|
885 |
assumes X: "distributed M MX X Px" |
|
886 |
and fin: "finite_measure MX" |
|
887 |
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
888 |
shows "entropy b MX X \<le> log b (measure MX (space MX))" |
|
889 |
proof - |
|
890 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
891 |
interpret finite_measure MX by fact |
|
892 |
have "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})" |
|
893 |
using int X by (intro entropy_le) auto |
|
894 |
also have "\<dots> \<le> log b (measure MX (space MX))" |
|
895 |
using Px distributed_imp_emeasure_nonzero[OF X] |
|
896 |
by (intro log_le) |
|
897 |
(auto intro!: borel_measurable_ereal_iff finite_measure_mono b_gt_1 |
|
898 |
less_le[THEN iffD2] measure_nonneg simp: emeasure_eq_measure) |
|
899 |
finally show ?thesis . |
|
900 |
qed |
|
901 |
||
47694 | 902 |
lemma (in information_space) entropy_uniform: |
49785 | 903 |
assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f") |
47694 | 904 |
shows "entropy b MX X = log b (measure MX A)" |
49785 | 905 |
proof (subst entropy_distr[OF X]) |
906 |
have [simp]: "emeasure MX A \<noteq> \<infinity>" |
|
907 |
using uniform_distributed_params[OF X] by (auto simp add: measure_def) |
|
908 |
have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) = |
|
909 |
(\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)" |
|
910 |
using measure_nonneg[of MX A] uniform_distributed_params[OF X] |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
911 |
by (intro integral_cong) (auto split: split_indicator simp: log_divide_eq) |
49785 | 912 |
show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) = |
913 |
log b (measure MX A)" |
|
914 |
unfolding eq using uniform_distributed_params[OF X] |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
915 |
by (subst integral_mult_right) (auto simp: measure_def) |
49785 | 916 |
qed |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
917 |
|
47694 | 918 |
lemma (in information_space) entropy_simple_distributed: |
49786 | 919 |
"simple_distributed M X f \<Longrightarrow> \<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))" |
920 |
by (subst entropy_distr[OF simple_distributed]) |
|
921 |
(auto simp add: lebesgue_integral_count_space_finite) |
|
39097 | 922 |
|
40859 | 923 |
lemma (in information_space) entropy_le_card_not_0: |
47694 | 924 |
assumes X: "simple_distributed M X f" |
925 |
shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))" |
|
39097 | 926 |
proof - |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
927 |
let ?X = "count_space (X`space M)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
928 |
have "\<H>(X) \<le> log b (measure ?X {x \<in> space ?X. f x \<noteq> 0})" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
929 |
by (rule entropy_le[OF simple_distributed[OF X]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
930 |
(simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
931 |
also have "measure ?X {x \<in> space ?X. f x \<noteq> 0} = card (X ` space M \<inter> {x. f x \<noteq> 0})" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
932 |
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def Int_def) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
933 |
finally show ?thesis . |
39097 | 934 |
qed |
935 |
||
40859 | 936 |
lemma (in information_space) entropy_le_card: |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
937 |
assumes X: "simple_distributed M X f" |
40859 | 938 |
shows "\<H>(X) \<le> log b (real (card (X ` space M)))" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
939 |
proof - |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
940 |
let ?X = "count_space (X`space M)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
941 |
have "\<H>(X) \<le> log b (measure ?X (space ?X))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
942 |
by (rule entropy_le_space[OF simple_distributed[OF X]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
943 |
(simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space finite_measure_count_space) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
944 |
also have "measure ?X (space ?X) = card (X ` space M)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
945 |
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def) |
39097 | 946 |
finally show ?thesis . |
947 |
qed |
|
948 |
||
949 |
subsection {* Conditional Mutual Information *} |
|
950 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
951 |
definition (in prob_space) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
952 |
"conditional_mutual_information b MX MY MZ X Y Z \<equiv> |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
953 |
mutual_information b MX (MY \<Otimes>\<^sub>M MZ) X (\<lambda>x. (Y x, Z x)) - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
954 |
mutual_information b MX MZ X Z" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
955 |
|
40859 | 956 |
abbreviation (in information_space) |
957 |
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where |
|
36624 | 958 |
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b |
47694 | 959 |
(count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
960 |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
961 |
lemma (in information_space) |
47694 | 962 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P" |
50003 | 963 |
assumes Px[measurable]: "distributed M S X Px" |
964 |
assumes Pz[measurable]: "distributed M P Z Pz" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
965 |
assumes Pyz[measurable]: "distributed M (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) Pyz" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
966 |
assumes Pxz[measurable]: "distributed M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) Pxz" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
967 |
assumes Pxyz[measurable]: "distributed M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
968 |
assumes I1: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
969 |
assumes I2: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
970 |
shows conditional_mutual_information_generic_eq: "conditional_mutual_information b S T P X Y Z |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
971 |
= (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" (is "?eq") |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
972 |
and conditional_mutual_information_generic_nonneg: "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg") |
40859 | 973 |
proof - |
47694 | 974 |
interpret S: sigma_finite_measure S by fact |
975 |
interpret T: sigma_finite_measure T by fact |
|
976 |
interpret P: sigma_finite_measure P by fact |
|
977 |
interpret TP: pair_sigma_finite T P .. |
|
978 |
interpret SP: pair_sigma_finite S P .. |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
979 |
interpret ST: pair_sigma_finite S T .. |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
980 |
interpret SPT: pair_sigma_finite "S \<Otimes>\<^sub>M P" T .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
981 |
interpret STP: pair_sigma_finite S "T \<Otimes>\<^sub>M P" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
982 |
interpret TPS: pair_sigma_finite "T \<Otimes>\<^sub>M P" S .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
983 |
have TP: "sigma_finite_measure (T \<Otimes>\<^sub>M P)" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
984 |
have SP: "sigma_finite_measure (S \<Otimes>\<^sub>M P)" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
985 |
have YZ: "random_variable (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x))" |
47694 | 986 |
using Pyz by (simp add: distributed_measurable) |
987 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
988 |
from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
989 |
distr (distr M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). (x, z))" |
50003 | 990 |
by (simp add: comp_def distr_distr) |
40859 | 991 |
|
47694 | 992 |
have "mutual_information b S P X Z = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
993 |
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^sub>M P))" |
47694 | 994 |
by (rule mutual_information_distr[OF S P Px Pz Pxz]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
995 |
also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" |
47694 | 996 |
using b_gt_1 Pxz Px Pz |
50003 | 997 |
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) (auto simp: split_beta') |
47694 | 998 |
finally have mi_eq: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
999 |
"mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" . |
47694 | 1000 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1001 |
have ae1: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0" |
47694 | 1002 |
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1003 |
moreover have ae2: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1004 |
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1005 |
moreover have ae3: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1006 |
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1007 |
moreover have ae4: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1008 |
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1009 |
moreover have ae5: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Px (fst x)" |
50003 | 1010 |
using Px by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1011 |
moreover have ae6: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pyz (snd x)" |
50003 | 1012 |
using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1013 |
moreover have ae7: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd (snd x))" |
50003 | 1014 |
using Pz Pz[THEN distributed_real_measurable] |
1015 |
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1016 |
moreover have ae8: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pxz (fst x, snd (snd x))" |
47694 | 1017 |
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair] |
50003 | 1018 |
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure) |
47694 | 1019 |
moreover note Pxyz[THEN distributed_real_AE] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1020 |
ultimately have ae: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. |
47694 | 1021 |
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) - |
1022 |
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) = |
|
1023 |
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) " |
|
1024 |
proof eventually_elim |
|
1025 |
case (goal1 x) |
|
1026 |
show ?case |
|
40859 | 1027 |
proof cases |
47694 | 1028 |
assume "Pxyz x \<noteq> 0" |
1029 |
with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x" |
|
1030 |
by auto |
|
1031 |
then show ?thesis |
|
56544 | 1032 |
using b_gt_1 by (simp add: log_simps less_imp_le field_simps) |
40859 | 1033 |
qed simp |
1034 |
qed |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1035 |
with I1 I2 show ?eq |
40859 | 1036 |
unfolding conditional_mutual_information_def |
47694 | 1037 |
apply (subst mi_eq) |
1038 |
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz]) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1039 |
apply (subst integral_diff[symmetric]) |
47694 | 1040 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
1041 |
done |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1042 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1043 |
let ?P = "density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) Pxyz" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1044 |
interpret P: prob_space ?P |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1045 |
unfolding distributed_distr_eq_density[OF Pxyz, symmetric] |
50003 | 1046 |
by (rule prob_space_distr) simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1047 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1048 |
let ?Q = "density (T \<Otimes>\<^sub>M P) Pyz" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1049 |
interpret Q: prob_space ?Q |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1050 |
unfolding distributed_distr_eq_density[OF Pyz, symmetric] |
50003 | 1051 |
by (rule prob_space_distr) simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1052 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1053 |
let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1054 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1055 |
from subdensity_real[of snd, OF _ Pyz Pz] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1056 |
have aeX1: "AE x in T \<Otimes>\<^sub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1057 |
have aeX2: "AE x in T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd x)" |
50003 | 1058 |
using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1059 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1060 |
have aeX3: "AE y in T \<Otimes>\<^sub>M P. (\<integral>\<^sup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))" |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1061 |
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz] |
50003 | 1062 |
by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1063 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1064 |
have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^sup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" |
56996 | 1065 |
apply (subst nn_integral_density) |
50003 | 1066 |
apply simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1067 |
apply (rule distributed_AE[OF Pxyz]) |
50003 | 1068 |
apply auto [] |
56996 | 1069 |
apply (rule nn_integral_mono_AE) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1070 |
using ae5 ae6 ae7 ae8 |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1071 |
apply eventually_elim |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1072 |
apply auto |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1073 |
done |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1074 |
also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)" |
56996 | 1075 |
by (subst STP.nn_integral_snd[symmetric]) (auto simp add: split_beta') |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1076 |
also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)" |
56996 | 1077 |
apply (rule nn_integral_cong_AE) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1078 |
using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1079 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1080 |
proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1081 |
fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1082 |
"(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1083 |
then show "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))" |
56996 | 1084 |
by (subst nn_integral_multc) |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1085 |
(auto split: prod.split) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1086 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1087 |
also have "\<dots> = 1" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1088 |
using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz] |
56996 | 1089 |
by (subst nn_integral_density[symmetric]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1090 |
finally have le1: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> 1" . |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1091 |
also have "\<dots> < \<infinity>" by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1092 |
finally have fin: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1093 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1094 |
have pos: "(\<integral>\<^sup>+x. ?f x \<partial>?P) \<noteq> 0" |
56996 | 1095 |
apply (subst nn_integral_density) |
50003 | 1096 |
apply simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1097 |
apply (rule distributed_AE[OF Pxyz]) |
50003 | 1098 |
apply auto [] |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1099 |
apply (simp add: split_beta') |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1100 |
proof |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1101 |
let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1102 |
assume "(\<integral>\<^sup>+x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1103 |
then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. ?g x \<le> 0" |
56996 | 1104 |
by (intro nn_integral_0_iff_AE[THEN iffD1]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1105 |
then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x = 0" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1106 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1107 |
by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1108 |
then have "(\<integral>\<^sup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) = 0" |
56996 | 1109 |
by (subst nn_integral_cong_AE[of _ "\<lambda>x. 0"]) auto |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1110 |
with P.emeasure_space_1 show False |
56996 | 1111 |
by (subst (asm) emeasure_density) (auto cong: nn_integral_cong) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1112 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1113 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1114 |
have neg: "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) = 0" |
56996 | 1115 |
apply (rule nn_integral_0_iff_AE[THEN iffD2]) |
50003 | 1116 |
apply simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1117 |
apply (subst AE_density) |
50003 | 1118 |
apply simp |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1119 |
using ae5 ae6 ae7 ae8 |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1120 |
apply eventually_elim |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1121 |
apply auto |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1122 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1123 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1124 |
have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1125 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integrable_diff[OF I1 I2]]) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1126 |
using ae |
50003 | 1127 |
apply (auto simp: split_beta') |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1128 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1129 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1130 |
have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1131 |
proof (intro le_imp_neg_le log_le[OF b_gt_1]) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1132 |
have If: "integrable ?P ?f" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1133 |
unfolding real_integrable_def |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1134 |
proof (intro conjI) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1135 |
from neg show "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1136 |
by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1137 |
from fin show "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1138 |
by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1139 |
qed simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1140 |
then have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) = (\<integral>x. ?f x \<partial>?P)" |
56996 | 1141 |
apply (rule nn_integral_eq_integral) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1142 |
apply (subst AE_density) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1143 |
apply simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1144 |
using ae5 ae6 ae7 ae8 |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1145 |
apply eventually_elim |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1146 |
apply auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1147 |
done |
56996 | 1148 |
with nn_integral_nonneg[of ?P ?f] pos le1 |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1149 |
show "0 < (\<integral>x. ?f x \<partial>?P)" "(\<integral>x. ?f x \<partial>?P) \<le> 1" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1150 |
by (simp_all add: one_ereal_def) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1151 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1152 |
also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)" |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1153 |
proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1154 |
show "AE x in ?P. ?f x \<in> {0<..}" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1155 |
unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1156 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
56544 | 1157 |
by eventually_elim (auto) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1158 |
show "integrable ?P ?f" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1159 |
unfolding real_integrable_def |
50003 | 1160 |
using fin neg by (auto simp: split_beta') |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1161 |
show "integrable ?P (\<lambda>x. - log b (?f x))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1162 |
apply (subst integrable_real_density) |
50003 | 1163 |
apply simp |
1164 |
apply (auto intro!: distributed_real_AE[OF Pxyz]) [] |
|
1165 |
apply simp |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1166 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3]) |
50003 | 1167 |
apply simp |
1168 |
apply simp |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1169 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1170 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1171 |
apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1172 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1173 |
qed (auto simp: b_gt_1 minus_log_convex) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1174 |
also have "\<dots> = conditional_mutual_information b S T P X Y Z" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1175 |
unfolding `?eq` |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1176 |
apply (subst integral_real_density) |
50003 | 1177 |
apply simp |
1178 |
apply (auto intro!: distributed_real_AE[OF Pxyz]) [] |
|
1179 |
apply simp |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1180 |
apply (intro integral_cong_AE) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1181 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1182 |
apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1183 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1184 |
finally show ?nonneg |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1185 |
by simp |
40859 | 1186 |
qed |
1187 |
||
49803 | 1188 |
lemma (in information_space) |
1189 |
fixes Px :: "_ \<Rightarrow> real" |
|
1190 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P" |
|
1191 |
assumes Fx: "finite_entropy S X Px" |
|
1192 |
assumes Fz: "finite_entropy P Z Pz" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1193 |
assumes Fyz: "finite_entropy (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) Pyz" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1194 |
assumes Fxz: "finite_entropy (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) Pxz" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1195 |
assumes Fxyz: "finite_entropy (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
49803 | 1196 |
shows conditional_mutual_information_generic_eq': "conditional_mutual_information b S T P X Y Z |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1197 |
= (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" (is "?eq") |
49803 | 1198 |
and conditional_mutual_information_generic_nonneg': "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg") |
1199 |
proof - |
|
50003 | 1200 |
note Px = Fx[THEN finite_entropy_distributed, measurable] |
1201 |
note Pz = Fz[THEN finite_entropy_distributed, measurable] |
|
1202 |
note Pyz = Fyz[THEN finite_entropy_distributed, measurable] |
|
1203 |
note Pxz = Fxz[THEN finite_entropy_distributed, measurable] |
|
1204 |
note Pxyz = Fxyz[THEN finite_entropy_distributed, measurable] |
|
49803 | 1205 |
|
1206 |
interpret S: sigma_finite_measure S by fact |
|
1207 |
interpret T: sigma_finite_measure T by fact |
|
1208 |
interpret P: sigma_finite_measure P by fact |
|
1209 |
interpret TP: pair_sigma_finite T P .. |
|
1210 |
interpret SP: pair_sigma_finite S P .. |
|
1211 |
interpret ST: pair_sigma_finite S T .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1212 |
interpret SPT: pair_sigma_finite "S \<Otimes>\<^sub>M P" T .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1213 |
interpret STP: pair_sigma_finite S "T \<Otimes>\<^sub>M P" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1214 |
interpret TPS: pair_sigma_finite "T \<Otimes>\<^sub>M P" S .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1215 |
have TP: "sigma_finite_measure (T \<Otimes>\<^sub>M P)" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1216 |
have SP: "sigma_finite_measure (S \<Otimes>\<^sub>M P)" .. |
49803 | 1217 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1218 |
from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1219 |
distr (distr M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). (x, z))" |
50003 | 1220 |
by (simp add: distr_distr comp_def) |
49803 | 1221 |
|
1222 |
have "mutual_information b S P X Z = |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1223 |
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^sub>M P))" |
49803 | 1224 |
by (rule mutual_information_distr[OF S P Px Pz Pxz]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1225 |
also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" |
49803 | 1226 |
using b_gt_1 Pxz Px Pz |
1227 |
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) |
|
50003 | 1228 |
(auto simp: split_beta') |
49803 | 1229 |
finally have mi_eq: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1230 |
"mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" . |
49803 | 1231 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1232 |
have ae1: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0" |
49803 | 1233 |
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1234 |
moreover have ae2: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1235 |
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1236 |
moreover have ae3: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1237 |
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1238 |
moreover have ae4: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
50003 | 1239 |
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1240 |
moreover have ae5: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Px (fst x)" |
50003 | 1241 |
using Px by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1242 |
moreover have ae6: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pyz (snd x)" |
50003 | 1243 |
using Pyz by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1244 |
moreover have ae7: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd (snd x))" |
50003 | 1245 |
using Pz Pz[THEN distributed_real_measurable] by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1246 |
moreover have ae8: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pxz (fst x, snd (snd x))" |
49803 | 1247 |
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair] |
1248 |
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def) |
|
1249 |
moreover note ae9 = Pxyz[THEN distributed_real_AE] |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1250 |
ultimately have ae: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. |
49803 | 1251 |
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) - |
1252 |
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) = |
|
1253 |
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) " |
|
1254 |
proof eventually_elim |
|
1255 |
case (goal1 x) |
|
1256 |
show ?case |
|
1257 |
proof cases |
|
1258 |
assume "Pxyz x \<noteq> 0" |
|
1259 |
with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x" |
|
1260 |
by auto |
|
1261 |
then show ?thesis |
|
56544 | 1262 |
using b_gt_1 by (simp add: log_simps less_imp_le field_simps) |
49803 | 1263 |
qed simp |
1264 |
qed |
|
1265 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1266 |
have "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) |
49803 | 1267 |
(\<lambda>x. Pxyz x * log b (Pxyz x) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pyz (snd x)))" |
1268 |
using finite_entropy_integrable[OF Fxyz] |
|
1269 |
using finite_entropy_integrable_transform[OF Fx Pxyz, of fst] |
|
1270 |
using finite_entropy_integrable_transform[OF Fyz Pxyz, of snd] |
|
1271 |
by simp |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1272 |
moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z)))) \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)" |
50003 | 1273 |
using Pxyz Px Pyz by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1274 |
ultimately have I1: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))" |
49803 | 1275 |
apply (rule integrable_cong_AE_imp) |
1276 |
using ae1 ae4 ae5 ae6 ae9 |
|
1277 |
by eventually_elim |
|
56536 | 1278 |
(auto simp: log_divide_eq log_mult_eq field_simps zero_less_mult_iff) |
49803 | 1279 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1280 |
have "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) |
49803 | 1281 |
(\<lambda>x. Pxyz x * log b (Pxz (fst x, snd (snd x))) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pz (snd (snd x))))" |
1282 |
using finite_entropy_integrable_transform[OF Fxz Pxyz, of "\<lambda>x. (fst x, snd (snd x))"] |
|
1283 |
using finite_entropy_integrable_transform[OF Fx Pxyz, of fst] |
|
1284 |
using finite_entropy_integrable_transform[OF Fz Pxyz, of "snd \<circ> snd"] |
|
50003 | 1285 |
by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1286 |
moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z))) \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)" |
49803 | 1287 |
using Pxyz Px Pz |
50003 | 1288 |
by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1289 |
ultimately have I2: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))" |
49803 | 1290 |
apply (rule integrable_cong_AE_imp) |
1291 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 ae9 |
|
1292 |
by eventually_elim |
|
56536 | 1293 |
(auto simp: log_divide_eq log_mult_eq field_simps zero_less_mult_iff) |
49803 | 1294 |
|
1295 |
from ae I1 I2 show ?eq |
|
1296 |
unfolding conditional_mutual_information_def |
|
1297 |
apply (subst mi_eq) |
|
1298 |
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz]) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1299 |
apply (subst integral_diff[symmetric]) |
49803 | 1300 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
1301 |
done |
|
1302 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1303 |
let ?P = "density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) Pxyz" |
49803 | 1304 |
interpret P: prob_space ?P |
50003 | 1305 |
unfolding distributed_distr_eq_density[OF Pxyz, symmetric] by (rule prob_space_distr) simp |
49803 | 1306 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1307 |
let ?Q = "density (T \<Otimes>\<^sub>M P) Pyz" |
49803 | 1308 |
interpret Q: prob_space ?Q |
50003 | 1309 |
unfolding distributed_distr_eq_density[OF Pyz, symmetric] by (rule prob_space_distr) simp |
49803 | 1310 |
|
1311 |
let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)" |
|
1312 |
||
1313 |
from subdensity_real[of snd, OF _ Pyz Pz] |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1314 |
have aeX1: "AE x in T \<Otimes>\<^sub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1315 |
have aeX2: "AE x in T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd x)" |
50003 | 1316 |
using Pz by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE) |
49803 | 1317 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1318 |
have aeX3: "AE y in T \<Otimes>\<^sub>M P. (\<integral>\<^sup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))" |
49803 | 1319 |
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz] |
50003 | 1320 |
by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1321 |
have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^sup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" |
56996 | 1322 |
apply (subst nn_integral_density) |
49803 | 1323 |
apply (rule distributed_borel_measurable[OF Pxyz]) |
1324 |
apply (rule distributed_AE[OF Pxyz]) |
|
50003 | 1325 |
apply simp |
56996 | 1326 |
apply (rule nn_integral_mono_AE) |
49803 | 1327 |
using ae5 ae6 ae7 ae8 |
1328 |
apply eventually_elim |
|
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1329 |
apply auto |
49803 | 1330 |
done |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1331 |
also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)" |
56996 | 1332 |
by (subst STP.nn_integral_snd[symmetric]) (auto simp add: split_beta') |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1333 |
also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)" |
56996 | 1334 |
apply (rule nn_integral_cong_AE) |
49803 | 1335 |
using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space |
1336 |
apply eventually_elim |
|
1337 |
proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure) |
|
1338 |
fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1339 |
"(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1340 |
then show "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))" |
56996 | 1341 |
by (subst nn_integral_multc) auto |
49803 | 1342 |
qed |
1343 |
also have "\<dots> = 1" |
|
1344 |
using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz] |
|
56996 | 1345 |
by (subst nn_integral_density[symmetric]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1346 |
finally have le1: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> 1" . |
49803 | 1347 |
also have "\<dots> < \<infinity>" by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1348 |
finally have fin: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp |
49803 | 1349 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1350 |
have pos: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> 0" |
56996 | 1351 |
apply (subst nn_integral_density) |
50003 | 1352 |
apply simp |
49803 | 1353 |
apply (rule distributed_AE[OF Pxyz]) |
50003 | 1354 |
apply simp |
49803 | 1355 |
apply (simp add: split_beta') |
1356 |
proof |
|
1357 |
let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1358 |
assume "(\<integral>\<^sup>+ x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1359 |
then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. ?g x \<le> 0" |
56996 | 1360 |
by (intro nn_integral_0_iff_AE[THEN iffD1]) (auto intro!: borel_measurable_ereal measurable_If) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1361 |
then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x = 0" |
49803 | 1362 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
1363 |
by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1364 |
then have "(\<integral>\<^sup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) = 0" |
56996 | 1365 |
by (subst nn_integral_cong_AE[of _ "\<lambda>x. 0"]) auto |
49803 | 1366 |
with P.emeasure_space_1 show False |
56996 | 1367 |
by (subst (asm) emeasure_density) (auto cong: nn_integral_cong) |
49803 | 1368 |
qed |
1369 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1370 |
have neg: "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) = 0" |
56996 | 1371 |
apply (rule nn_integral_0_iff_AE[THEN iffD2]) |
50003 | 1372 |
apply (auto simp: split_beta') [] |
49803 | 1373 |
apply (subst AE_density) |
50003 | 1374 |
apply (auto simp: split_beta') [] |
49803 | 1375 |
using ae5 ae6 ae7 ae8 |
1376 |
apply eventually_elim |
|
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1377 |
apply auto |
49803 | 1378 |
done |
1379 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1380 |
have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1381 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integrable_diff[OF I1 I2]]) |
49803 | 1382 |
using ae |
50003 | 1383 |
apply (auto simp: split_beta') |
49803 | 1384 |
done |
1385 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1386 |
have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)" |
49803 | 1387 |
proof (intro le_imp_neg_le log_le[OF b_gt_1]) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1388 |
have If: "integrable ?P ?f" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1389 |
unfolding real_integrable_def |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1390 |
proof (intro conjI) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1391 |
from neg show "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1392 |
by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1393 |
from fin show "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1394 |
by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1395 |
qed simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1396 |
then have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) = (\<integral>x. ?f x \<partial>?P)" |
56996 | 1397 |
apply (rule nn_integral_eq_integral) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1398 |
apply (subst AE_density) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1399 |
apply simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1400 |
using ae5 ae6 ae7 ae8 |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1401 |
apply eventually_elim |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1402 |
apply auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1403 |
done |
56996 | 1404 |
with nn_integral_nonneg[of ?P ?f] pos le1 |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1405 |
show "0 < (\<integral>x. ?f x \<partial>?P)" "(\<integral>x. ?f x \<partial>?P) \<le> 1" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1406 |
by (simp_all add: one_ereal_def) |
49803 | 1407 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1408 |
also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)" |
49803 | 1409 |
proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"]) |
1410 |
show "AE x in ?P. ?f x \<in> {0<..}" |
|
1411 |
unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]] |
|
1412 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
|
56544 | 1413 |
by eventually_elim (auto) |
49803 | 1414 |
show "integrable ?P ?f" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1415 |
unfolding real_integrable_def |
50003 | 1416 |
using fin neg by (auto simp: split_beta') |
49803 | 1417 |
show "integrable ?P (\<lambda>x. - log b (?f x))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1418 |
apply (subst integrable_real_density) |
50003 | 1419 |
apply simp |
1420 |
apply (auto intro!: distributed_real_AE[OF Pxyz]) [] |
|
1421 |
apply simp |
|
49803 | 1422 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3]) |
50003 | 1423 |
apply simp |
1424 |
apply simp |
|
49803 | 1425 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
1426 |
apply eventually_elim |
|
1427 |
apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps) |
|
1428 |
done |
|
1429 |
qed (auto simp: b_gt_1 minus_log_convex) |
|
1430 |
also have "\<dots> = conditional_mutual_information b S T P X Y Z" |
|
1431 |
unfolding `?eq` |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1432 |
apply (subst integral_real_density) |
50003 | 1433 |
apply simp |
1434 |
apply (auto intro!: distributed_real_AE[OF Pxyz]) [] |
|
1435 |
apply simp |
|
49803 | 1436 |
apply (intro integral_cong_AE) |
1437 |
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE] |
|
1438 |
apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps) |
|
1439 |
done |
|
1440 |
finally show ?nonneg |
|
1441 |
by simp |
|
1442 |
qed |
|
1443 |
||
40859 | 1444 |
lemma (in information_space) conditional_mutual_information_eq: |
47694 | 1445 |
assumes Pz: "simple_distributed M Z Pz" |
1446 |
assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz" |
|
1447 |
assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz" |
|
1448 |
assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
1449 |
shows "\<I>(X ; Y | Z) = |
|
1450 |
(\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x))`space M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))" |
|
1451 |
proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _ |
|
1452 |
simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz] |
|
1453 |
simple_distributed_joint2[OF Pxyz]]) |
|
1454 |
note simple_distributed_joint2_finite[OF Pxyz, simp] |
|
1455 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
1456 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1457 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
1458 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1459 |
show "sigma_finite_measure (count_space (Z ` space M))" |
|
1460 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1461 |
have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M) = |
47694 | 1462 |
count_space (X`space M \<times> Y`space M \<times> Z`space M)" |
1463 |
(is "?P = ?C") |
|
1464 |
by (simp add: pair_measure_count_space) |
|
40859 | 1465 |
|
47694 | 1466 |
let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1467 |
have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Z ` space M))" |
47694 | 1468 |
using simple_distributed_joint[OF Pxz] by (rule distributed_measurable) |
1469 |
from measurable_comp[OF this measurable_fst] |
|
1470 |
have "random_variable (count_space (X ` space M)) X" |
|
1471 |
by (simp add: comp_def) |
|
1472 |
then have "simple_function M X" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49999
diff
changeset
|
1473 |
unfolding simple_function_def by (auto simp: measurable_count_space_eq2) |
47694 | 1474 |
then have "simple_distributed M X ?Px" |
1475 |
by (rule simple_distributedI) auto |
|
1476 |
then show "distributed M (count_space (X ` space M)) X ?Px" |
|
1477 |
by (rule simple_distributed) |
|
1478 |
||
1479 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)" |
|
1480 |
let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)" |
|
1481 |
let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)" |
|
1482 |
show |
|
1483 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))" |
|
1484 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))" |
|
1485 |
by (auto intro!: integrable_count_space simp: pair_measure_count_space) |
|
1486 |
let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))" |
|
1487 |
let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))" |
|
1488 |
have "(\<lambda>(x, y, z). ?i x y z) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then ?j (fst x) (fst (snd x)) (snd (snd x)) else 0)" |
|
1489 |
by (auto intro!: ext) |
|
1490 |
then show "(\<integral> (x, y, z). ?i x y z \<partial>?P) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. ?j x y z)" |
|
57418 | 1491 |
by (auto intro!: setsum.cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum.If_cases split_beta') |
36624 | 1492 |
qed |
1493 |
||
47694 | 1494 |
lemma (in information_space) conditional_mutual_information_nonneg: |
1495 |
assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z" |
|
1496 |
shows "0 \<le> \<I>(X ; Y | Z)" |
|
1497 |
proof - |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1498 |
have [simp]: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M) = |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1499 |
count_space (X`space M \<times> Y`space M \<times> Z`space M)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1500 |
by (simp add: pair_measure_count_space X Y Z simple_functionD) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1501 |
note sf = sigma_finite_measure_count_space_finite[OF simple_functionD(1)] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1502 |
note sd = simple_distributedI[OF _ refl] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1503 |
note sp = simple_function_Pair |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1504 |
show ?thesis |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1505 |
apply (rule conditional_mutual_information_generic_nonneg[OF sf[OF X] sf[OF Y] sf[OF Z]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1506 |
apply (rule simple_distributed[OF sd[OF X]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1507 |
apply (rule simple_distributed[OF sd[OF Z]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1508 |
apply (rule simple_distributed_joint[OF sd[OF sp[OF Y Z]]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1509 |
apply (rule simple_distributed_joint[OF sd[OF sp[OF X Z]]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1510 |
apply (rule simple_distributed_joint2[OF sd[OF sp[OF X sp[OF Y Z]]]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1511 |
apply (auto intro!: integrable_count_space simp: X Y Z simple_functionD) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1512 |
done |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1513 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1514 |
|
39097 | 1515 |
subsection {* Conditional Entropy *} |
1516 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1517 |
definition (in prob_space) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1518 |
"conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1519 |
real (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1520 |
|
40859 | 1521 |
abbreviation (in information_space) |
1522 |
conditional_entropy_Pow ("\<H>'(_ | _')") where |
|
47694 | 1523 |
"\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1524 |
|
49791 | 1525 |
lemma (in information_space) conditional_entropy_generic_eq: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1526 |
fixes Pxy :: "_ \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
49791 | 1527 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
50003 | 1528 |
assumes Py[measurable]: "distributed M T Y Py" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1529 |
assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1530 |
shows "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^sub>M T))" |
49791 | 1531 |
proof - |
1532 |
interpret S: sigma_finite_measure S by fact |
|
1533 |
interpret T: sigma_finite_measure T by fact |
|
1534 |
interpret ST: pair_sigma_finite S T .. |
|
1535 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1536 |
have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) x)" |
49791 | 1537 |
unfolding AE_density[OF distributed_borel_measurable, OF Pxy] |
1538 |
unfolding distributed_distr_eq_density[OF Pxy] |
|
1539 |
using distributed_RN_deriv[OF Pxy] |
|
1540 |
by auto |
|
1541 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1542 |
have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real (RN_deriv T (distr M T Y) (snd x))" |
49791 | 1543 |
unfolding AE_density[OF distributed_borel_measurable, OF Pxy] |
1544 |
unfolding distributed_distr_eq_density[OF Py] |
|
1545 |
apply (rule ST.AE_pair_measure) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1546 |
apply auto |
49791 | 1547 |
using distributed_RN_deriv[OF Py] |
1548 |
apply auto |
|
1549 |
done |
|
1550 |
ultimately |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1551 |
have "conditional_entropy b S T X Y = - (\<integral>x. Pxy x * log b (Pxy x / Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))" |
49791 | 1552 |
unfolding conditional_entropy_def neg_equal_iff_equal |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1553 |
apply (subst integral_real_density[symmetric]) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1554 |
apply (auto simp: distributed_real_AE[OF Pxy] distributed_distr_eq_density[OF Pxy] |
49791 | 1555 |
intro!: integral_cong_AE) |
1556 |
done |
|
1557 |
then show ?thesis by (simp add: split_beta') |
|
1558 |
qed |
|
1559 |
||
1560 |
lemma (in information_space) conditional_entropy_eq_entropy: |
|
47694 | 1561 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
1562 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1563 |
assumes Py[measurable]: "distributed M T Y Py" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1564 |
assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1565 |
assumes I1: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1566 |
assumes I2: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1567 |
shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y" |
40859 | 1568 |
proof - |
47694 | 1569 |
interpret S: sigma_finite_measure S by fact |
1570 |
interpret T: sigma_finite_measure T by fact |
|
1571 |
interpret ST: pair_sigma_finite S T .. |
|
1572 |
||
1573 |
have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)" |
|
49786 | 1574 |
by (rule entropy_distr[OF Py]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1575 |
also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^sub>M T))" |
47694 | 1576 |
using b_gt_1 Py[THEN distributed_real_measurable] |
1577 |
by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1578 |
finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^sub>M T))" . |
49791 | 1579 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1580 |
have ae2: "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 1581 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1582 |
moreover have ae4: "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Py (snd x)" |
47694 | 1583 |
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1584 |
moreover note ae5 = Pxy[THEN distributed_real_AE] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1585 |
ultimately have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Pxy x \<and> 0 \<le> Py (snd x) \<and> |
49790
6b9b9ebba47d
remove unneeded assumption from conditional_entropy_generic_eq
hoelzl
parents:
49788
diff
changeset
|
1586 |
(Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Py (snd x)))" |
47694 | 1587 |
by eventually_elim auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1588 |
then have ae: "AE x in S \<Otimes>\<^sub>M T. |
47694 | 1589 |
Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))" |
56544 | 1590 |
by eventually_elim (auto simp: log_simps field_simps b_gt_1) |
49791 | 1591 |
have "conditional_entropy b S T X Y = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1592 |
- (\<integral>x. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))" |
49791 | 1593 |
unfolding conditional_entropy_generic_eq[OF S T Py Pxy] neg_equal_iff_equal |
1594 |
apply (intro integral_cong_AE) |
|
1595 |
using ae |
|
1596 |
apply auto |
|
47694 | 1597 |
done |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1598 |
also have "\<dots> = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^sub>M T)) - - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))" |
49791 | 1599 |
by (simp add: integral_diff[OF I1 I2]) |
1600 |
finally show ?thesis |
|
1601 |
unfolding conditional_entropy_generic_eq[OF S T Py Pxy] entropy_distr[OF Pxy] e_eq |
|
1602 |
by (simp add: split_beta') |
|
1603 |
qed |
|
1604 |
||
1605 |
lemma (in information_space) conditional_entropy_eq_entropy_simple: |
|
1606 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1607 |
shows "\<H>(X | Y) = entropy b (count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x)) - \<H>(Y)" |
49791 | 1608 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1609 |
have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)" |
49791 | 1610 |
(is "?P = ?C") using X Y by (simp add: simple_functionD pair_measure_count_space) |
1611 |
show ?thesis |
|
1612 |
by (rule conditional_entropy_eq_entropy sigma_finite_measure_count_space_finite |
|
1613 |
simple_functionD X Y simple_distributed simple_distributedI[OF _ refl] |
|
1614 |
simple_distributed_joint simple_function_Pair integrable_count_space)+ |
|
1615 |
(auto simp: `?P = ?C` intro!: integrable_count_space simple_functionD X Y) |
|
39097 | 1616 |
qed |
1617 |
||
40859 | 1618 |
lemma (in information_space) conditional_entropy_eq: |
49792
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1619 |
assumes Y: "simple_distributed M Y Py" |
47694 | 1620 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
1621 |
shows "\<H>(X | Y) = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))" |
|
1622 |
proof (subst conditional_entropy_generic_eq[OF _ _ |
|
49790
6b9b9ebba47d
remove unneeded assumption from conditional_entropy_generic_eq
hoelzl
parents:
49788
diff
changeset
|
1623 |
simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
49792
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1624 |
have "finite ((\<lambda>x. (X x, Y x))`space M)" |
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1625 |
using XY unfolding simple_distributed_def by auto |
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1626 |
from finite_imageI[OF this, of fst] |
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1627 |
have [simp]: "finite (X`space M)" |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
53374
diff
changeset
|
1628 |
by (simp add: image_comp comp_def) |
47694 | 1629 |
note Y[THEN simple_distributed_finite, simp] |
1630 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
1631 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1632 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
1633 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1634 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1635 |
have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)" |
47694 | 1636 |
(is "?P = ?C") |
49792
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1637 |
using Y by (simp add: simple_distributed_finite pair_measure_count_space) |
47694 | 1638 |
have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) = |
1639 |
(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / Py (snd x)) else 0)" |
|
1640 |
by auto |
|
49792
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1641 |
from Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) = |
47694 | 1642 |
- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))" |
57418 | 1643 |
by (auto intro!: setsum.cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum.If_cases split_beta') |
47694 | 1644 |
qed |
39097 | 1645 |
|
47694 | 1646 |
lemma (in information_space) conditional_mutual_information_eq_conditional_entropy: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1647 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
47694 | 1648 |
shows "\<I>(X ; X | Y) = \<H>(X | Y)" |
1649 |
proof - |
|
1650 |
def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0" |
|
1651 |
def Pxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M) else 0" |
|
1652 |
def Pxxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, X x, Y x))`space M then measure M ((\<lambda>x. (X x, X x, Y x)) -` {x} \<inter> space M) else 0" |
|
1653 |
let ?M = "X`space M \<times> X`space M \<times> Y`space M" |
|
39097 | 1654 |
|
47694 | 1655 |
note XY = simple_function_Pair[OF X Y] |
1656 |
note XXY = simple_function_Pair[OF X XY] |
|
1657 |
have Py: "simple_distributed M Y Py" |
|
1658 |
using Y by (rule simple_distributedI) (auto simp: Py_def) |
|
1659 |
have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
1660 |
using XY by (rule simple_distributedI) (auto simp: Pxy_def) |
|
1661 |
have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy" |
|
1662 |
using XXY by (rule simple_distributedI) (auto simp: Pxxy_def) |
|
1663 |
have eq: "(\<lambda>x. (X x, X x, Y x)) ` space M = (\<lambda>(x, y). (x, x, y)) ` (\<lambda>x. (X x, Y x)) ` space M" |
|
1664 |
by auto |
|
1665 |
have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A" |
|
1666 |
by (auto simp: inj_on_def) |
|
1667 |
have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)" |
|
1668 |
by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob]) |
|
1669 |
have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1670 |
by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair) |
|
1671 |
then show ?thesis |
|
1672 |
apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy]) |
|
49792
43f49922811d
remove unnecessary assumption from conditional_entropy_eq
hoelzl
parents:
49791
diff
changeset
|
1673 |
apply (subst conditional_entropy_eq[OF Py Pxy]) |
57418 | 1674 |
apply (auto intro!: setsum.cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum.reindex[OF inj] |
47694 | 1675 |
log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space) |
1676 |
using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE] |
|
49790
6b9b9ebba47d
remove unneeded assumption from conditional_entropy_generic_eq
hoelzl
parents:
49788
diff
changeset
|
1677 |
apply (auto simp add: not_le[symmetric] AE_count_space) |
47694 | 1678 |
done |
1679 |
qed |
|
1680 |
||
1681 |
lemma (in information_space) conditional_entropy_nonneg: |
|
1682 |
assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)" |
|
1683 |
using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y] |
|
1684 |
by simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1685 |
|
39097 | 1686 |
subsection {* Equalities *} |
1687 |
||
47694 | 1688 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr: |
1689 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real" |
|
1690 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1691 |
assumes Px[measurable]: "distributed M S X Px" and Py[measurable]: "distributed M T Y Py" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1692 |
assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1693 |
assumes Ix: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1694 |
assumes Iy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1695 |
assumes Ixy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1696 |
shows "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" |
40859 | 1697 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1698 |
have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^sub>M T))" |
47694 | 1699 |
using b_gt_1 Px[THEN distributed_real_measurable] |
49786 | 1700 |
apply (subst entropy_distr[OF Px]) |
47694 | 1701 |
apply (subst distributed_transform_integral[OF Pxy Px, where T=fst]) |
1702 |
apply (auto intro!: integral_cong) |
|
1703 |
done |
|
1704 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1705 |
have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))" |
47694 | 1706 |
using b_gt_1 Py[THEN distributed_real_measurable] |
49786 | 1707 |
apply (subst entropy_distr[OF Py]) |
47694 | 1708 |
apply (subst distributed_transform_integral[OF Pxy Py, where T=snd]) |
1709 |
apply (auto intro!: integral_cong) |
|
1710 |
done |
|
1711 |
||
1712 |
interpret S: sigma_finite_measure S by fact |
|
1713 |
interpret T: sigma_finite_measure T by fact |
|
1714 |
interpret ST: pair_sigma_finite S T .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1715 |
have ST: "sigma_finite_measure (S \<Otimes>\<^sub>M T)" .. |
47694 | 1716 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1717 |
have XY: "entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^sub>M T))" |
49786 | 1718 |
by (subst entropy_distr[OF Pxy]) (auto intro!: integral_cong) |
47694 | 1719 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1720 |
have "AE x in S \<Otimes>\<^sub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 1721 |
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1722 |
moreover have "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 1723 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1724 |
moreover have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Px (fst x)" |
47694 | 1725 |
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1726 |
moreover have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Py (snd x)" |
47694 | 1727 |
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
1728 |
moreover note Pxy[THEN distributed_real_AE] |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1729 |
ultimately have "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) = |
47694 | 1730 |
Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
1731 |
(is "AE x in _. ?f x = ?g x") |
|
1732 |
proof eventually_elim |
|
1733 |
case (goal1 x) |
|
1734 |
show ?case |
|
1735 |
proof cases |
|
1736 |
assume "Pxy x \<noteq> 0" |
|
1737 |
with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x" |
|
1738 |
by auto |
|
1739 |
then show ?thesis |
|
56544 | 1740 |
using b_gt_1 by (simp add: log_simps less_imp_le field_simps) |
47694 | 1741 |
qed simp |
1742 |
qed |
|
1743 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1744 |
have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?f" |
47694 | 1745 |
unfolding X Y XY |
1746 |
apply (subst integral_diff) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1747 |
apply (intro integrable_diff Ixy Ix Iy)+ |
47694 | 1748 |
apply (subst integral_diff) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1749 |
apply (intro Ixy Ix Iy)+ |
47694 | 1750 |
apply (simp add: field_simps) |
1751 |
done |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1752 |
also have "\<dots> = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?g" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56571
diff
changeset
|
1753 |
using `AE x in _. ?f x = ?g x` by (intro integral_cong_AE) auto |
47694 | 1754 |
also have "\<dots> = mutual_information b S T X Y" |
1755 |
by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric]) |
|
1756 |
finally show ?thesis .. |
|
1757 |
qed |
|
1758 |
||
49802
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1759 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy': |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1760 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real" |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1761 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1762 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1763 |
assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1764 |
assumes Ix: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1765 |
assumes Iy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1766 |
assumes Ixy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
49802
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1767 |
shows "mutual_information b S T X Y = entropy b S X - conditional_entropy b S T X Y" |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1768 |
using |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1769 |
mutual_information_eq_entropy_conditional_entropy_distr[OF S T Px Py Pxy Ix Iy Ixy] |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1770 |
conditional_entropy_eq_entropy[OF S T Py Pxy Ixy Iy] |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1771 |
by simp |
dd8dffaf84b9
continuous version of mutual_information_eq_entropy_conditional_entropy
hoelzl
parents:
49792
diff
changeset
|
1772 |
|
47694 | 1773 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy: |
1774 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1775 |
shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)" |
|
1776 |
proof - |
|
1777 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1778 |
using sf_X by (rule simple_distributedI) auto |
|
1779 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1780 |
using sf_Y by (rule simple_distributedI) auto |
|
1781 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1782 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1783 |
then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))" |
|
1784 |
by (rule simple_distributedI) auto |
|
1785 |
from simple_distributed_joint_finite[OF this, simp] |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1786 |
have eq: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
47694 | 1787 |
by (simp add: pair_measure_count_space) |
1788 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1789 |
have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))" |
47694 | 1790 |
using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY] |
1791 |
by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space) |
|
1792 |
then show ?thesis |
|
49791 | 1793 |
unfolding conditional_entropy_eq_entropy_simple[OF sf_X sf_Y] by simp |
47694 | 1794 |
qed |
1795 |
||
1796 |
lemma (in information_space) mutual_information_nonneg_simple: |
|
1797 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1798 |
shows "0 \<le> \<I>(X ; Y)" |
|
1799 |
proof - |
|
1800 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1801 |
using sf_X by (rule simple_distributedI) auto |
|
1802 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1803 |
using sf_Y by (rule simple_distributedI) auto |
|
1804 |
||
1805 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1806 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1807 |
then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))" |
|
1808 |
by (rule simple_distributedI) auto |
|
1809 |
||
1810 |
from simple_distributed_joint_finite[OF this, simp] |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1811 |
have eq: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
47694 | 1812 |
by (simp add: pair_measure_count_space) |
1813 |
||
40859 | 1814 |
show ?thesis |
47694 | 1815 |
by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
1816 |
(simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite) |
|
40859 | 1817 |
qed |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1818 |
|
40859 | 1819 |
lemma (in information_space) conditional_entropy_less_eq_entropy: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1820 |
assumes X: "simple_function M X" and Z: "simple_function M Z" |
40859 | 1821 |
shows "\<H>(X | Z) \<le> \<H>(X)" |
36624 | 1822 |
proof - |
47694 | 1823 |
have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple) |
1824 |
also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] . |
|
1825 |
finally show ?thesis by auto |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1826 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1827 |
|
49803 | 1828 |
lemma (in information_space) |
1829 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real" |
|
1830 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1831 |
assumes Px: "finite_entropy S X Px" and Py: "finite_entropy T Y Py" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1832 |
assumes Pxy: "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
49803 | 1833 |
shows "conditional_entropy b S T X Y \<le> entropy b S X" |
1834 |
proof - |
|
1835 |
||
1836 |
have "0 \<le> mutual_information b S T X Y" |
|
1837 |
by (rule mutual_information_nonneg') fact+ |
|
1838 |
also have "\<dots> = entropy b S X - conditional_entropy b S T X Y" |
|
1839 |
apply (rule mutual_information_eq_entropy_conditional_entropy') |
|
1840 |
using assms |
|
1841 |
by (auto intro!: finite_entropy_integrable finite_entropy_distributed |
|
1842 |
finite_entropy_integrable_transform[OF Px] |
|
1843 |
finite_entropy_integrable_transform[OF Py]) |
|
1844 |
finally show ?thesis by auto |
|
1845 |
qed |
|
1846 |
||
40859 | 1847 |
lemma (in information_space) entropy_chain_rule: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1848 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
40859 | 1849 |
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)" |
1850 |
proof - |
|
47694 | 1851 |
note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl] |
1852 |
note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl] |
|
1853 |
note simple_distributed_joint_finite[OF this, simp] |
|
1854 |
let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" |
|
1855 |
let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)" |
|
1856 |
let ?h = "\<lambda>x. if x \<in> (\<lambda>x. (Y x, X x)) ` space M then prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M) else 0" |
|
1857 |
have "\<H>(\<lambda>x. (X x, Y x)) = - (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` space M. ?f x * log b (?f x))" |
|
1858 |
using XY by (rule entropy_simple_distributed) |
|
1859 |
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>(x, y). (y, x)) ` (\<lambda>x. (X x, Y x)) ` space M. ?g x * log b (?g x))" |
|
57418 | 1860 |
by (subst (2) setsum.reindex) (auto simp: inj_on_def intro!: setsum.cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"]) |
47694 | 1861 |
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))" |
57418 | 1862 |
by (auto intro!: setsum.cong) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1863 |
also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^sub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" |
49786 | 1864 |
by (subst entropy_distr[OF simple_distributed_joint[OF YX]]) |
47694 | 1865 |
(auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite |
57418 | 1866 |
cong del: setsum.cong intro!: setsum.mono_neutral_left) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50419
diff
changeset
|
1867 |
finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^sub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" . |
47694 | 1868 |
then show ?thesis |
49791 | 1869 |
unfolding conditional_entropy_eq_entropy_simple[OF Y X] by simp |
36624 | 1870 |
qed |
1871 |
||
40859 | 1872 |
lemma (in information_space) entropy_partition: |
47694 | 1873 |
assumes X: "simple_function M X" |
1874 |
shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)" |
|
36624 | 1875 |
proof - |
47694 | 1876 |
note fX = simple_function_compose[OF X, of f] |
1877 |
have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto |
|
1878 |
have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A" |
|
1879 |
by (auto simp: inj_on_def) |
|
1880 |
show ?thesis |
|
1881 |
apply (subst entropy_chain_rule[symmetric, OF fX X]) |
|
1882 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]]) |
|
1883 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1884 |
unfolding eq |
|
57418 | 1885 |
apply (subst setsum.reindex[OF inj]) |
1886 |
apply (auto intro!: setsum.cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"]) |
|
47694 | 1887 |
done |
36624 | 1888 |
qed |
1889 |
||
40859 | 1890 |
corollary (in information_space) entropy_data_processing: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1891 |
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)" |
40859 | 1892 |
proof - |
47694 | 1893 |
note fX = simple_function_compose[OF X, of f] |
1894 |
from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition) |
|
40859 | 1895 |
then show "\<H>(f \<circ> X) \<le> \<H>(X)" |
47694 | 1896 |
by (auto intro: conditional_entropy_nonneg[OF X fX]) |
40859 | 1897 |
qed |
36624 | 1898 |
|
40859 | 1899 |
corollary (in information_space) entropy_of_inj: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1900 |
assumes X: "simple_function M X" and inj: "inj_on f (X`space M)" |
36624 | 1901 |
shows "\<H>(f \<circ> X) = \<H>(X)" |
1902 |
proof (rule antisym) |
|
40859 | 1903 |
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] . |
36624 | 1904 |
next |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1905 |
have sf: "simple_function M (f \<circ> X)" |
40859 | 1906 |
using X by auto |
36624 | 1907 |
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))" |
47694 | 1908 |
unfolding o_assoc |
1909 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1910 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"]) |
|
57418 | 1911 |
apply (auto intro!: setsum.cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def) |
47694 | 1912 |
done |
36624 | 1913 |
also have "... \<le> \<H>(f \<circ> X)" |
40859 | 1914 |
using entropy_data_processing[OF sf] . |
36624 | 1915 |
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" . |
1916 |
qed |
|
1917 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1918 |
end |