author | hoelzl |
Wed, 10 Oct 2012 12:12:26 +0200 | |
changeset 49788 | 3c10763f5cb4 |
parent 49787 | d8de705b48d4 |
child 49790 | 6b9b9ebba47d |
permissions | -rw-r--r-- |
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(* 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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Added Information theory and Example: dining cryptographers
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theory Information |
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imports |
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Independent_Family |
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move conditional expectation to its own theory file
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Radon_Nikodym |
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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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section "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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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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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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lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq |
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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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"entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N" |
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definition |
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"KL_divergence b M N = integral\<^isup>L N (entropy_density b M N)" |
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lemma (in information_space) measurable_entropy_density: |
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assumes ac: "absolutely_continuous M N" "sets N = events" |
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shows "entropy_density b M N \<in> borel_measurable M" |
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proof - |
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from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis |
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unfolding entropy_density_def |
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by (intro measurable_comp) auto |
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qed |
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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: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
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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_density) |
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show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M" |
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using f |
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by (auto simp: comp_def entropy_density_def intro!: borel_measurable_log borel_measurable_RN_deriv_density) |
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have "density M (RN_deriv M (density M f)) = density M f" |
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using f by (intro density_RN_deriv_density) auto |
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then have eq: "AE x in M. RN_deriv M (density M f) x = f x" |
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using f |
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by (intro density_unique) |
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(auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg) |
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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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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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lemma (in sigma_finite_measure) KL_density_density: |
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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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using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density divide_nonneg_nonneg) |
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also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)" |
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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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lemma (in information_space) KL_gt_0: |
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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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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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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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have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0" |
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using D by (subst positive_integral_0_iff_AE) auto |
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154 |
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have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)" |
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using D by (simp add: emeasure_density cong: positive_integral_cong) |
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then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1" |
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using N.emeasure_space_1 by simp |
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have "integrable M D" "integral\<^isup>L M D = 1" |
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using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all |
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have "0 \<le> 1 - measure M ?D_set" |
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using prob_le_1 by (auto simp: field_simps) |
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also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)" |
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using `integrable M D` `integral\<^isup>L M D = 1` |
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by (simp add: emeasure_eq_measure) |
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also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)" |
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proof (rule integral_less_AE) |
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show "integrable M (\<lambda>x. D x - indicator ?D_set x)" |
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using `integrable M D` |
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by (intro integral_diff integral_indicator) auto |
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next |
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from integral_cmult(1)[OF int, of "ln b"] |
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show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" |
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by (simp add: ac_simps) |
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next |
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show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0" |
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proof |
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assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0" |
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then have disj: "AE x in M. D x = 1 \<or> D x = 0" |
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using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect) |
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183 |
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have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)" |
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185 |
using D(1) by auto |
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also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) \<partial>M)" |
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using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def) |
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finally have "AE x in M. D x = 1" |
189 |
using D D_pos by (intro AE_I_eq_1) auto |
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then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)" |
191 |
by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric]) |
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47694 | 192 |
also have "\<dots> = density M D A" |
193 |
using `A \<in> sets M` D by (simp add: emeasure_density) |
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finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp |
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qed |
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196 |
show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M" |
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using D(1) by (auto intro: sets_Collect_conj) |
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198 |
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47694 | 199 |
show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow> |
200 |
D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))" |
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201 |
using D(2) |
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proof (eventually_elim, safe) |
203 |
fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t" |
|
204 |
and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))" |
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205 |
|
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206 |
have "D t - 1 = D t - indicator ?D_set t" |
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207 |
using Dt by simp |
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208 |
also note eq |
47694 | 209 |
also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)" |
210 |
using b_gt_1 `D t \<noteq> 0` `0 \<le> D t` |
|
211 |
by (simp add: log_def ln_div less_le) |
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212 |
finally have "ln (1 / D t) = 1 / D t - 1" |
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213 |
using `D t \<noteq> 0` by (auto simp: field_simps) |
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214 |
from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1` |
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215 |
show False by auto |
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216 |
qed |
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217 |
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47694 | 218 |
show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
219 |
using D(2) AE_space |
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220 |
proof eventually_elim |
|
221 |
fix t assume "t \<in> space M" "0 \<le> D t" |
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222 |
show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
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223 |
proof cases |
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224 |
assume asm: "D t \<noteq> 0" |
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225 |
then have "0 < D t" using `0 \<le> D t` by auto |
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226 |
then have "0 < 1 / D t" by auto |
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changeset
|
227 |
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
|
228 |
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
|
229 |
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
|
230 |
using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto |
47694 | 231 |
also have "\<dots> = D t * (ln b * log b (D t))" |
232 |
using `0 < D t` b_gt_1 |
|
233 |
by (simp_all add: log_def ln_div) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
234 |
finally show ?thesis by simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
235 |
qed simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
236 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
237 |
qed |
47694 | 238 |
also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)" |
239 |
by (simp add: ac_simps) |
|
240 |
also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)" |
|
241 |
using int by (rule integral_cmult) |
|
242 |
finally show ?thesis |
|
243 |
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
|
244 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
245 |
|
47694 | 246 |
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
|
247 |
proof - |
47694 | 248 |
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
|
249 |
proof (rule RN_deriv_unique) |
47694 | 250 |
show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto |
251 |
show "density M (\<lambda>x. 1) = M" |
|
252 |
apply (auto intro!: measure_eqI emeasure_density) |
|
253 |
apply (subst emeasure_density) |
|
254 |
apply auto |
|
255 |
done |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
256 |
qed |
47694 | 257 |
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
|
258 |
by (elim AE_mp) simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
259 |
from integral_cong_AE[OF this] |
47694 | 260 |
have "integral\<^isup>L M (entropy_density b M M) = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
261 |
by (simp add: entropy_density_def comp_def) |
47694 | 262 |
then show "KL_divergence b M M = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
263 |
unfolding KL_divergence_def |
47694 | 264 |
by auto |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
265 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
266 |
|
47694 | 267 |
lemma (in information_space) KL_eq_0_iff_eq: |
268 |
fixes D :: "'a \<Rightarrow> real" |
|
269 |
assumes "prob_space (density M D)" |
|
270 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
271 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
272 |
shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M" |
|
273 |
using KL_same_eq_0[of b] KL_gt_0[OF assms] |
|
274 |
by (auto simp: less_le) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
275 |
|
47694 | 276 |
lemma (in information_space) KL_eq_0_iff_eq_ac: |
277 |
fixes D :: "'a \<Rightarrow> real" |
|
278 |
assumes "prob_space N" |
|
279 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
|
280 |
assumes int: "integrable N (entropy_density b M N)" |
|
281 |
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
|
282 |
proof - |
47694 | 283 |
interpret N: prob_space N by fact |
284 |
have "finite_measure N" by unfold_locales |
|
285 |
from real_RN_deriv[OF this ac] guess D . note D = this |
|
286 |
||
287 |
have "N = density M (RN_deriv M N)" |
|
288 |
using ac by (rule density_RN_deriv[symmetric]) |
|
289 |
also have "\<dots> = density M D" |
|
290 |
using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong) |
|
291 |
finally have N: "N = density M D" . |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
292 |
|
47694 | 293 |
from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density |
294 |
have "integrable N (\<lambda>x. log b (D x))" |
|
295 |
by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int]) |
|
296 |
(auto simp: N entropy_density_def) |
|
297 |
with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))" |
|
298 |
by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def) |
|
299 |
with `prob_space N` D show ?thesis |
|
300 |
unfolding N |
|
301 |
by (intro KL_eq_0_iff_eq) auto |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
302 |
qed |
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
303 |
|
47694 | 304 |
lemma (in information_space) KL_nonneg: |
305 |
assumes "prob_space (density M D)" |
|
306 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
307 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
308 |
shows "0 \<le> KL_divergence b M (density M D)" |
|
309 |
using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0) |
|
40859 | 310 |
|
47694 | 311 |
lemma (in sigma_finite_measure) KL_density_density_nonneg: |
312 |
fixes f g :: "'a \<Rightarrow> real" |
|
313 |
assumes "1 < b" |
|
314 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)" |
|
315 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)" |
|
316 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
317 |
assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))" |
|
318 |
shows "0 \<le> KL_divergence b (density M f) (density M g)" |
|
319 |
proof - |
|
320 |
interpret Mf: prob_space "density M f" by fact |
|
321 |
interpret Mf: information_space "density M f" b by default fact |
|
322 |
have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _") |
|
323 |
using f g ac by (subst density_density_divide) simp_all |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
324 |
|
47694 | 325 |
have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))" |
326 |
proof (rule Mf.KL_nonneg) |
|
327 |
show "prob_space ?DD" unfolding eq by fact |
|
328 |
from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)" |
|
329 |
by auto |
|
330 |
show "AE x in density M f. 0 \<le> g x / f x" |
|
331 |
using f g by (auto simp: AE_density divide_nonneg_nonneg) |
|
332 |
show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))" |
|
333 |
using `1 < b` f g ac |
|
334 |
by (subst integral_density) |
|
335 |
(auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If) |
|
336 |
qed |
|
337 |
also have "\<dots> = KL_divergence b (density M f) (density M g)" |
|
338 |
using f g ac by (subst density_density_divide) simp_all |
|
339 |
finally show ?thesis . |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
340 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
341 |
|
39097 | 342 |
subsection {* Mutual Information *} |
343 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
344 |
definition (in prob_space) |
38656 | 345 |
"mutual_information b S T X Y = |
47694 | 346 |
KL_divergence b (distr M S X \<Otimes>\<^isub>M distr M T Y) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
347 |
|
47694 | 348 |
lemma (in information_space) mutual_information_indep_vars: |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
349 |
fixes S T X Y |
47694 | 350 |
defines "P \<equiv> distr M S X \<Otimes>\<^isub>M distr M T Y" |
351 |
defines "Q \<equiv> distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
352 |
shows "indep_var S X T Y \<longleftrightarrow> |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
353 |
(random_variable S X \<and> random_variable T Y \<and> |
47694 | 354 |
absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and> |
355 |
mutual_information b S T X Y = 0)" |
|
356 |
unfolding indep_var_distribution_eq |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
357 |
proof safe |
47694 | 358 |
assume rv: "random_variable S X" "random_variable T Y" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
359 |
|
47694 | 360 |
interpret X: prob_space "distr M S X" |
361 |
by (rule prob_space_distr) fact |
|
362 |
interpret Y: prob_space "distr M T Y" |
|
363 |
by (rule prob_space_distr) fact |
|
364 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default |
|
365 |
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
|
366 |
|
47694 | 367 |
interpret Q: prob_space Q unfolding Q_def |
368 |
by (rule prob_space_distr) (simp add: comp_def measurable_pair_iff rv) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
369 |
|
47694 | 370 |
{ assume "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
371 |
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
|
372 |
|
47694 | 373 |
show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def) |
374 |
then have ed: "entropy_density b P Q \<in> borel_measurable P" |
|
375 |
by (rule P.measurable_entropy_density) simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
376 |
|
47694 | 377 |
have "AE x in P. 1 = RN_deriv P Q x" |
378 |
proof (rule P.RN_deriv_unique) |
|
379 |
show "density P (\<lambda>x. 1) = Q" |
|
380 |
unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density) |
|
381 |
qed auto |
|
382 |
then have ae_0: "AE x in P. entropy_density b P Q x = 0" |
|
383 |
by eventually_elim (auto simp: entropy_density_def) |
|
384 |
then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)" |
|
385 |
using ed unfolding `Q = P` by (intro integrable_cong_AE) auto |
|
386 |
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
|
387 |
|
47694 | 388 |
show "mutual_information b S T X Y = 0" |
389 |
unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P` |
|
390 |
using ae_0 by (simp cong: integral_cong_AE) } |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
391 |
|
47694 | 392 |
{ assume ac: "absolutely_continuous P Q" |
393 |
assume int: "integrable Q (entropy_density b P Q)" |
|
394 |
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
|
395 |
|
47694 | 396 |
have eq: "Q = P" |
397 |
proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1]) |
|
398 |
show "prob_space Q" by unfold_locales |
|
399 |
show "absolutely_continuous P Q" by fact |
|
400 |
show "integrable Q (entropy_density b P Q)" by fact |
|
401 |
show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure) |
|
402 |
show "KL_divergence b P Q = 0" |
|
403 |
using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def) |
|
404 |
qed |
|
405 |
then show "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
406 |
unfolding P_def Q_def .. } |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
407 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
408 |
|
40859 | 409 |
abbreviation (in information_space) |
410 |
mutual_information_Pow ("\<I>'(_ ; _')") where |
|
47694 | 411 |
"\<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
|
412 |
|
47694 | 413 |
lemma (in information_space) |
414 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
415 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
|
416 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
417 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
418 |
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
|
419 |
shows mutual_information_distr: "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R") |
|
420 |
and mutual_information_nonneg: "integrable (S \<Otimes>\<^isub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y" |
|
40859 | 421 |
proof - |
47694 | 422 |
have X: "random_variable S X" |
423 |
using Px by (auto simp: distributed_def) |
|
424 |
have Y: "random_variable T Y" |
|
425 |
using Py by (auto simp: distributed_def) |
|
426 |
interpret S: sigma_finite_measure S by fact |
|
427 |
interpret T: sigma_finite_measure T by fact |
|
428 |
interpret ST: pair_sigma_finite S T .. |
|
429 |
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr) |
|
430 |
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr) |
|
431 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" .. |
|
432 |
let ?P = "S \<Otimes>\<^isub>M T" |
|
433 |
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))" |
|
434 |
||
435 |
{ fix A assume "A \<in> sets S" |
|
436 |
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)" |
|
437 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
438 |
intro!: arg_cong[where f="emeasure M"]) } |
|
439 |
note marginal_eq1 = this |
|
440 |
{ fix A assume "A \<in> sets T" |
|
441 |
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)" |
|
442 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
443 |
intro!: arg_cong[where f="emeasure M"]) } |
|
444 |
note marginal_eq2 = this |
|
445 |
||
446 |
have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))" |
|
447 |
by auto |
|
448 |
||
449 |
have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))" |
|
450 |
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq |
|
451 |
proof (subst pair_measure_density) |
|
452 |
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T" |
|
453 |
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)" |
|
454 |
using Px Py by (auto simp: distributed_def) |
|
455 |
show "sigma_finite_measure (density S Px)" unfolding Px(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
456 |
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
457 |
qed (fact | simp)+ |
|
458 |
||
459 |
have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))" |
|
460 |
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] .. |
|
461 |
||
462 |
from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P" |
|
463 |
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'') |
|
464 |
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)" |
|
465 |
proof (rule ST.AE_pair_measure) |
|
466 |
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P" |
|
467 |
using f by auto |
|
468 |
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))" |
|
469 |
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE) |
|
470 |
qed |
|
471 |
||
472 |
have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)" |
|
473 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
474 |
moreover |
|
475 |
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)" |
|
476 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
477 |
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
478 |
by eventually_elim auto |
|
479 |
||
480 |
show "?M = ?R" |
|
481 |
unfolding M f_def |
|
482 |
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac |
|
483 |
by (rule ST.KL_density_density) |
|
484 |
||
485 |
assume int: "integrable (S \<Otimes>\<^isub>M T) f" |
|
486 |
show "0 \<le> ?M" unfolding M |
|
487 |
proof (rule ST.KL_density_density_nonneg |
|
488 |
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]]) |
|
489 |
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) " |
|
490 |
unfolding distributed_distr_eq_density[OF Pxy, symmetric] |
|
491 |
using distributed_measurable[OF Pxy] by (rule prob_space_distr) |
|
492 |
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))" |
|
493 |
unfolding distr_eq[symmetric] by unfold_locales |
|
40859 | 494 |
qed |
495 |
qed |
|
496 |
||
497 |
lemma (in information_space) |
|
47694 | 498 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
499 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
|
500 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
501 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
502 |
assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y" |
|
503 |
shows mutual_information_eq_0: "mutual_information b S T X Y = 0" |
|
36624 | 504 |
proof - |
47694 | 505 |
interpret S: sigma_finite_measure S by fact |
506 |
interpret T: sigma_finite_measure T by fact |
|
507 |
interpret ST: pair_sigma_finite S T .. |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
508 |
|
47694 | 509 |
have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
510 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
511 |
moreover |
|
512 |
have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
513 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
514 |
moreover |
|
515 |
have "AE x in S \<Otimes>\<^isub>M T. Pxy x = Px (fst x) * Py (snd x)" |
|
516 |
using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy] |
|
517 |
by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'') |
|
518 |
ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0" |
|
519 |
by eventually_elim simp |
|
520 |
then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^isub>M T))" |
|
521 |
by (rule integral_cong_AE) |
|
522 |
then show ?thesis |
|
523 |
by (subst mutual_information_distr[OF assms(1-5)]) simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
524 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
525 |
|
47694 | 526 |
lemma (in information_space) mutual_information_simple_distributed: |
527 |
assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py" |
|
528 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
529 |
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)))" |
|
530 |
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
|
531 |
note fin = simple_distributed_joint_finite[OF XY, simp] |
|
532 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
533 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
534 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
535 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
536 |
let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
537 |
let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))" |
|
538 |
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)" |
|
539 |
by auto |
|
540 |
with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M))) = |
|
541 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))" |
|
542 |
by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta' |
|
543 |
intro!: setsum_cong) |
|
544 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
545 |
|
47694 | 546 |
lemma (in information_space) |
547 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
548 |
assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py" |
|
549 |
assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
550 |
assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)" |
|
551 |
shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0" |
|
552 |
proof (subst mutual_information_simple_distributed[OF Px Py Pxy]) |
|
553 |
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))) = |
|
554 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)" |
|
555 |
by (intro setsum_cong) (auto simp: ae) |
|
556 |
then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. |
|
557 |
Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp |
|
558 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
559 |
|
39097 | 560 |
subsection {* Entropy *} |
561 |
||
47694 | 562 |
definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where |
563 |
"entropy b S X = - KL_divergence b S (distr M S X)" |
|
564 |
||
40859 | 565 |
abbreviation (in information_space) |
566 |
entropy_Pow ("\<H>'(_')") where |
|
47694 | 567 |
"\<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
|
568 |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
569 |
lemma (in information_space) |
47694 | 570 |
fixes X :: "'a \<Rightarrow> 'b" |
49785 | 571 |
assumes X: "distributed M MX X f" |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
572 |
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
|
573 |
proof - |
49785 | 574 |
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X] |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
575 |
interpret X: prob_space "distr M MX X" |
49785 | 576 |
using D(1) by (rule prob_space_distr) |
577 |
||
578 |
have sf: "sigma_finite_measure (distr M MX X)" by default |
|
579 |
have ae: "AE x in MX. f x = RN_deriv MX (density MX f) x" |
|
580 |
using D |
|
581 |
by (intro RN_deriv_unique_sigma_finite) |
|
582 |
(auto intro: divide_nonneg_nonneg measure_nonneg |
|
583 |
simp: distributed_distr_eq_density[symmetric, OF X] sf) |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
584 |
|
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
585 |
have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) = |
49785 | 586 |
log b (f x)" |
587 |
unfolding distributed_distr_eq_density[OF X] |
|
588 |
apply (subst AE_density) |
|
589 |
using D apply simp |
|
590 |
using ae apply eventually_elim |
|
591 |
apply (subst (asm) eq_commute) |
|
592 |
apply auto |
|
593 |
done |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
594 |
|
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
595 |
have int_eq: "- (\<integral> x. log b (f x) \<partial>distr M MX X) = - (\<integral> x. f x * log b (f x) \<partial>MX)" |
49785 | 596 |
unfolding distributed_distr_eq_density[OF X] |
597 |
using D |
|
598 |
by (subst integral_density) |
|
599 |
(auto simp: borel_measurable_ereal_iff) |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
600 |
|
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
601 |
show ?eq |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
602 |
unfolding entropy_def KL_divergence_def entropy_density_def comp_def |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
603 |
apply (subst integral_cong_AE) |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
604 |
apply (rule ae_eq) |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
605 |
apply (rule int_eq) |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
606 |
done |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
607 |
qed |
49785 | 608 |
|
49786 | 609 |
lemma (in prob_space) distributed_imp_emeasure_nonzero: |
610 |
assumes X: "distributed M MX X Px" |
|
611 |
shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0" |
|
612 |
proof |
|
613 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
614 |
interpret X: prob_space "distr M MX X" |
|
615 |
using distributed_measurable[OF X] by (rule prob_space_distr) |
|
616 |
||
617 |
assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0" |
|
618 |
with Px have "AE x in MX. Px x = 0" |
|
619 |
by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ereal_iff) |
|
620 |
moreover |
|
621 |
from X.emeasure_space_1 have "(\<integral>\<^isup>+x. Px x \<partial>MX) = 1" |
|
622 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
623 |
by (subst (asm) emeasure_density) |
|
624 |
(auto simp: borel_measurable_ereal_iff intro!: integral_cong cong: positive_integral_cong) |
|
625 |
ultimately show False |
|
626 |
by (simp add: positive_integral_cong_AE) |
|
627 |
qed |
|
628 |
||
629 |
lemma (in information_space) entropy_le: |
|
630 |
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure" |
|
631 |
assumes X: "distributed M MX X Px" |
|
632 |
and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>" |
|
633 |
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
634 |
shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})" |
|
635 |
proof - |
|
636 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
637 |
interpret X: prob_space "distr M MX X" |
|
638 |
using distributed_measurable[OF X] by (rule prob_space_distr) |
|
639 |
||
640 |
have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) = |
|
641 |
- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)" |
|
642 |
using Px fin |
|
643 |
by (subst integral_indicator) (auto simp: measure_def borel_measurable_ereal_iff) |
|
644 |
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)" |
|
645 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
646 |
apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus]) |
|
647 |
by (subst integral_density) (auto simp: borel_measurable_ereal_iff intro!: integral_cong) |
|
648 |
also have "\<dots> \<le> (\<integral> x. - log b (1 / Px x) \<partial>distr M MX X)" |
|
649 |
proof (rule X.jensens_inequality[of "\<lambda>x. 1 / Px x" "{0<..}" 0 1 "\<lambda>x. - log b x"]) |
|
650 |
show "AE x in distr M MX X. 1 / Px x \<in> {0<..}" |
|
651 |
unfolding distributed_distr_eq_density[OF X] |
|
652 |
using Px by (auto simp: AE_density) |
|
653 |
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" |
|
654 |
by (auto simp: one_ereal_def) |
|
655 |
have "(\<integral>\<^isup>+ x. max 0 (ereal (- (if Px x = 0 then 0 else 1))) \<partial>MX) = (\<integral>\<^isup>+ x. 0 \<partial>MX)" |
|
656 |
by (intro positive_integral_cong) (auto split: split_max) |
|
657 |
then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)" |
|
658 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
659 |
by (auto simp: positive_integral_density integrable_def borel_measurable_ereal_iff fin positive_integral_max_0 |
|
660 |
cong: positive_integral_cong) |
|
661 |
have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) = |
|
662 |
integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
663 |
using Px |
|
664 |
by (intro integrable_cong_AE) |
|
665 |
(auto simp: borel_measurable_ereal_iff log_divide_eq |
|
666 |
intro!: measurable_If) |
|
667 |
then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))" |
|
668 |
unfolding distributed_distr_eq_density[OF X] |
|
669 |
using Px int |
|
670 |
by (subst integral_density) (auto simp: borel_measurable_ereal_iff) |
|
671 |
qed (auto simp: minus_log_convex[OF b_gt_1]) |
|
672 |
also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)" |
|
673 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
674 |
by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq) |
|
675 |
also have "\<dots> = - entropy b MX X" |
|
676 |
unfolding distributed_distr_eq_density[OF X] using Px |
|
677 |
by (subst entropy_distr[OF X]) (auto simp: borel_measurable_ereal_iff integral_density) |
|
678 |
finally show ?thesis |
|
679 |
by simp |
|
680 |
qed |
|
681 |
||
682 |
lemma (in information_space) entropy_le_space: |
|
683 |
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure" |
|
684 |
assumes X: "distributed M MX X Px" |
|
685 |
and fin: "finite_measure MX" |
|
686 |
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))" |
|
687 |
shows "entropy b MX X \<le> log b (measure MX (space MX))" |
|
688 |
proof - |
|
689 |
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X] |
|
690 |
interpret finite_measure MX by fact |
|
691 |
have "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})" |
|
692 |
using int X by (intro entropy_le) auto |
|
693 |
also have "\<dots> \<le> log b (measure MX (space MX))" |
|
694 |
using Px distributed_imp_emeasure_nonzero[OF X] |
|
695 |
by (intro log_le) |
|
696 |
(auto intro!: borel_measurable_ereal_iff finite_measure_mono b_gt_1 |
|
697 |
less_le[THEN iffD2] measure_nonneg simp: emeasure_eq_measure) |
|
698 |
finally show ?thesis . |
|
699 |
qed |
|
700 |
||
49785 | 701 |
lemma (in prob_space) uniform_distributed_params: |
702 |
assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" |
|
703 |
shows "A \<in> sets MX" "measure MX A \<noteq> 0" |
|
47694 | 704 |
proof - |
49785 | 705 |
interpret X: prob_space "distr M MX X" |
706 |
using distributed_measurable[OF X] by (rule prob_space_distr) |
|
707 |
||
708 |
show "measure MX A \<noteq> 0" |
|
709 |
proof |
|
710 |
assume "measure MX A = 0" |
|
711 |
with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X] |
|
712 |
show False |
|
713 |
by (simp add: emeasure_density zero_ereal_def[symmetric]) |
|
714 |
qed |
|
715 |
with measure_notin_sets[of A MX] show "A \<in> sets MX" |
|
716 |
by blast |
|
39097 | 717 |
qed |
36624 | 718 |
|
47694 | 719 |
lemma (in information_space) entropy_uniform: |
49785 | 720 |
assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f") |
47694 | 721 |
shows "entropy b MX X = log b (measure MX A)" |
49785 | 722 |
proof (subst entropy_distr[OF X]) |
723 |
have [simp]: "emeasure MX A \<noteq> \<infinity>" |
|
724 |
using uniform_distributed_params[OF X] by (auto simp add: measure_def) |
|
725 |
have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) = |
|
726 |
(\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)" |
|
727 |
using measure_nonneg[of MX A] uniform_distributed_params[OF X] |
|
728 |
by (auto intro!: integral_cong split: split_indicator simp: log_divide_eq) |
|
729 |
show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) = |
|
730 |
log b (measure MX A)" |
|
731 |
unfolding eq using uniform_distributed_params[OF X] |
|
732 |
by (subst lebesgue_integral_cmult) (auto simp: measure_def) |
|
733 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
734 |
|
47694 | 735 |
lemma (in information_space) entropy_simple_distributed: |
49786 | 736 |
"simple_distributed M X f \<Longrightarrow> \<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))" |
737 |
by (subst entropy_distr[OF simple_distributed]) |
|
738 |
(auto simp add: lebesgue_integral_count_space_finite) |
|
39097 | 739 |
|
40859 | 740 |
lemma (in information_space) entropy_le_card_not_0: |
47694 | 741 |
assumes X: "simple_distributed M X f" |
742 |
shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))" |
|
39097 | 743 |
proof - |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
744 |
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
|
745 |
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
|
746 |
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
|
747 |
(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
|
748 |
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
|
749 |
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
|
750 |
finally show ?thesis . |
39097 | 751 |
qed |
752 |
||
40859 | 753 |
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
|
754 |
assumes X: "simple_distributed M X f" |
40859 | 755 |
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
|
756 |
proof - |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
757 |
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
|
758 |
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
|
759 |
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
|
760 |
(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
|
761 |
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
|
762 |
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def) |
39097 | 763 |
finally show ?thesis . |
764 |
qed |
|
765 |
||
766 |
subsection {* Conditional Mutual Information *} |
|
767 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
768 |
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
|
769 |
"conditional_mutual_information b MX MY MZ X Y Z \<equiv> |
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
|
770 |
mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) - |
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
|
771 |
mutual_information b MX MZ X Z" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
772 |
|
40859 | 773 |
abbreviation (in information_space) |
774 |
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where |
|
36624 | 775 |
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b |
47694 | 776 |
(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
|
777 |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
778 |
lemma (in pair_sigma_finite) borel_measurable_positive_integral_fst: |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
779 |
"(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M2) \<in> borel_measurable M1" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
780 |
using positive_integral_fst_measurable(1)[of "\<lambda>(x, y). f x y"] by simp |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
781 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
782 |
lemma (in pair_sigma_finite) borel_measurable_positive_integral_snd: |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
783 |
assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (M2 \<Otimes>\<^isub>M M1)" shows "(\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M1) \<in> borel_measurable M2" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
784 |
proof - |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
785 |
interpret Q: pair_sigma_finite M2 M1 by default |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
786 |
from Q.borel_measurable_positive_integral_fst assms show ?thesis by simp |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
787 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
788 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
789 |
lemma (in information_space) |
47694 | 790 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P" |
791 |
assumes Px: "distributed M S X Px" |
|
792 |
assumes Pz: "distributed M P Z Pz" |
|
793 |
assumes Pyz: "distributed M (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz" |
|
794 |
assumes Pxz: "distributed M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz" |
|
795 |
assumes Pxyz: "distributed M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
796 |
assumes I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))" |
|
797 |
assumes I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>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
|
798 |
shows conditional_mutual_information_generic_eq: "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
|
799 |
= (\<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>\<^isub>M T \<Otimes>\<^isub>M P))" (is "?eq") |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
800 |
and conditional_mutual_information_generic_nonneg: "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg") |
40859 | 801 |
proof - |
47694 | 802 |
interpret S: sigma_finite_measure S by fact |
803 |
interpret T: sigma_finite_measure T by fact |
|
804 |
interpret P: sigma_finite_measure P by fact |
|
805 |
interpret TP: pair_sigma_finite T P .. |
|
806 |
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
|
807 |
interpret ST: pair_sigma_finite S T .. |
47694 | 808 |
interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T .. |
809 |
interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" .. |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
810 |
interpret TPS: pair_sigma_finite "T \<Otimes>\<^isub>M P" S .. |
47694 | 811 |
have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" .. |
812 |
have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" .. |
|
813 |
have YZ: "random_variable (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x))" |
|
814 |
using Pyz by (simp add: distributed_measurable) |
|
815 |
||
816 |
have Pxyz_f: "\<And>M f. f \<in> measurable M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) \<Longrightarrow> (\<lambda>x. Pxyz (f x)) \<in> borel_measurable M" |
|
817 |
using measurable_comp[OF _ Pxyz[THEN distributed_real_measurable]] by (auto simp: comp_def) |
|
818 |
||
819 |
{ fix f g h M |
|
820 |
assume f: "f \<in> measurable M S" and g: "g \<in> measurable M P" and h: "h \<in> measurable M (S \<Otimes>\<^isub>M P)" |
|
821 |
from measurable_comp[OF h Pxz[THEN distributed_real_measurable]] |
|
822 |
measurable_comp[OF f Px[THEN distributed_real_measurable]] |
|
823 |
measurable_comp[OF g Pz[THEN distributed_real_measurable]] |
|
824 |
have "(\<lambda>x. log b (Pxz (h x) / (Px (f x) * Pz (g x)))) \<in> borel_measurable M" |
|
825 |
by (simp add: comp_def b_gt_1) } |
|
826 |
note borel_log = this |
|
827 |
||
828 |
have measurable_cut: "(\<lambda>(x, y, z). (x, z)) \<in> measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (S \<Otimes>\<^isub>M P)" |
|
829 |
by (auto simp add: split_beta' comp_def intro!: measurable_Pair measurable_snd') |
|
830 |
||
831 |
from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) = |
|
832 |
distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))" |
|
833 |
by (subst distr_distr[OF measurable_cut]) (auto dest: distributed_measurable simp: comp_def) |
|
40859 | 834 |
|
47694 | 835 |
have "mutual_information b S P X Z = |
836 |
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))" |
|
837 |
by (rule mutual_information_distr[OF S P Px Pz Pxz]) |
|
838 |
also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" |
|
839 |
using b_gt_1 Pxz Px Pz |
|
840 |
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) |
|
841 |
(auto simp: split_beta' intro!: measurable_Pair measurable_snd' measurable_snd'' measurable_fst'' borel_measurable_times |
|
842 |
dest!: distributed_real_measurable) |
|
843 |
finally have mi_eq: |
|
844 |
"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>\<^isub>M T \<Otimes>\<^isub>M P))" . |
|
845 |
||
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
846 |
have ae1: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0" |
47694 | 847 |
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
848 |
moreover have ae2: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
47694 | 849 |
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) (auto intro: measurable_snd') |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
850 |
moreover have ae3: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
47694 | 851 |
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) (auto intro: measurable_Pair measurable_snd') |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
852 |
moreover have ae4: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
47694 | 853 |
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) (auto intro: measurable_Pair) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
854 |
moreover have ae5: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)" |
47694 | 855 |
using Px by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
856 |
moreover have ae6: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)" |
47694 | 857 |
using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
858 |
moreover have ae7: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))" |
47694 | 859 |
using Pz Pz[THEN distributed_real_measurable] by (auto intro!: measurable_snd'' TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] 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
|
860 |
moreover have ae8: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))" |
47694 | 861 |
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair] |
862 |
using measurable_comp[OF measurable_Pair[OF measurable_fst measurable_comp[OF measurable_snd measurable_snd]] Pxz[THEN distributed_real_measurable], of T] |
|
863 |
using measurable_comp[OF measurable_snd measurable_Pair2[OF Pxz[THEN distributed_real_measurable]], of _ T] |
|
864 |
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def) |
|
865 |
moreover note Pxyz[THEN distributed_real_AE] |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
866 |
ultimately have ae: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. |
47694 | 867 |
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) - |
868 |
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) = |
|
869 |
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) " |
|
870 |
proof eventually_elim |
|
871 |
case (goal1 x) |
|
872 |
show ?case |
|
40859 | 873 |
proof cases |
47694 | 874 |
assume "Pxyz x \<noteq> 0" |
875 |
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" |
|
876 |
by auto |
|
877 |
then show ?thesis |
|
878 |
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps) |
|
40859 | 879 |
qed simp |
880 |
qed |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
881 |
with I1 I2 show ?eq |
40859 | 882 |
unfolding conditional_mutual_information_def |
47694 | 883 |
apply (subst mi_eq) |
884 |
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz]) |
|
885 |
apply (subst integral_diff(2)[symmetric]) |
|
886 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
|
887 |
done |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
888 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
889 |
let ?P = "density (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) Pxyz" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
890 |
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
|
891 |
unfolding distributed_distr_eq_density[OF Pxyz, symmetric] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
892 |
using distributed_measurable[OF Pxyz] by (rule prob_space_distr) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
893 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
894 |
let ?Q = "density (T \<Otimes>\<^isub>M P) Pyz" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
895 |
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
|
896 |
unfolding distributed_distr_eq_density[OF Pyz, symmetric] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
897 |
using distributed_measurable[OF Pyz] by (rule prob_space_distr) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
898 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
899 |
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
|
900 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
901 |
from subdensity_real[of snd, OF _ Pyz Pz] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
902 |
have aeX1: "AE x in T \<Otimes>\<^isub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
903 |
have aeX2: "AE x in T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd x)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
904 |
using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
905 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
906 |
have aeX3: "AE y in T \<Otimes>\<^isub>M P. (\<integral>\<^isup>+ 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
|
907 |
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz] |
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
908 |
apply (intro TP.AE_pair_measure) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
909 |
apply (auto simp: comp_def measurable_split_conv |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
910 |
intro!: measurable_snd'' borel_measurable_ereal_eq borel_measurable_ereal |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
911 |
SP.borel_measurable_positive_integral_snd measurable_compose[OF _ Pxz[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
912 |
measurable_Pair |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
913 |
dest: distributed_real_AE distributed_real_measurable) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
914 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
915 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
916 |
note M = borel_measurable_divide borel_measurable_diff borel_measurable_times borel_measurable_ereal |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
917 |
measurable_compose[OF _ measurable_snd] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
918 |
measurable_Pair |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
919 |
measurable_compose[OF _ Pxyz[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
920 |
measurable_compose[OF _ Pxz[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
921 |
measurable_compose[OF _ Pyz[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
922 |
measurable_compose[OF _ Pz[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
923 |
measurable_compose[OF _ Px[THEN distributed_real_measurable]] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
924 |
STP.borel_measurable_positive_integral_snd |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
925 |
have "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^isup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
926 |
apply (subst positive_integral_density) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
927 |
apply (rule 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
|
928 |
apply (rule distributed_AE[OF Pxyz]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
929 |
apply (auto simp add: borel_measurable_ereal_iff split_beta' intro!: M) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
930 |
apply (rule positive_integral_mono_AE) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
931 |
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
|
932 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
933 |
apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
934 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
935 |
also have "\<dots> = (\<integral>\<^isup>+(y, z). \<integral>\<^isup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^isub>M P)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
936 |
by (subst STP.positive_integral_snd_measurable[symmetric]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
937 |
(auto simp add: borel_measurable_ereal_iff split_beta' intro!: M) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
938 |
also have "\<dots> = (\<integral>\<^isup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^isub>M P)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
939 |
apply (rule positive_integral_cong_AE) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
940 |
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
|
941 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
942 |
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
|
943 |
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" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
944 |
"(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
945 |
then show "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
946 |
apply (subst positive_integral_multc) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
947 |
apply (auto intro!: borel_measurable_ereal divide_nonneg_nonneg |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
948 |
measurable_compose[OF _ Pxz[THEN distributed_real_measurable]] measurable_Pair |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
949 |
split: prod.split) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
950 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
951 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
952 |
also have "\<dots> = 1" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
953 |
using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
954 |
by (subst positive_integral_density[symmetric]) (auto intro!: M) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
955 |
finally have le1: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> 1" . |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
956 |
also have "\<dots> < \<infinity>" by simp |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
957 |
finally have fin: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
958 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
959 |
have pos: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
960 |
apply (subst positive_integral_density) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
961 |
apply (rule 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
|
962 |
apply (rule distributed_AE[OF Pxyz]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
963 |
apply (auto simp add: borel_measurable_ereal_iff split_beta' intro!: M) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
964 |
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
|
965 |
proof |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
966 |
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)))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
967 |
assume "(\<integral>\<^isup>+ x. ?g x \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)) = 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
968 |
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. ?g x \<le> 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
969 |
by (intro positive_integral_0_iff_AE[THEN iffD1]) (auto intro!: M borel_measurable_ereal measurable_If) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
970 |
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxyz x = 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
971 |
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
|
972 |
by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
973 |
then have "(\<integral>\<^isup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) = 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
974 |
by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
975 |
with P.emeasure_space_1 show False |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
976 |
by (subst (asm) emeasure_density) (auto intro!: M cong: positive_integral_cong) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
977 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
978 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
979 |
have neg: "(\<integral>\<^isup>+ x. - ?f x \<partial>?P) = 0" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
980 |
apply (rule positive_integral_0_iff_AE[THEN iffD2]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
981 |
apply (auto intro!: M simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
982 |
apply (subst AE_density) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
983 |
apply (auto intro!: M simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
984 |
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
|
985 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
986 |
apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
987 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
988 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
989 |
have I3: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
990 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
991 |
using ae |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
992 |
apply (auto intro!: M simp: split_beta') |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
993 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
994 |
|
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
995 |
have "- log b 1 \<le> - log b (integral\<^isup>L ?P ?f)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
996 |
proof (intro le_imp_neg_le log_le[OF b_gt_1]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
997 |
show "0 < integral\<^isup>L ?P ?f" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
998 |
using neg pos fin positive_integral_positive[of ?P ?f] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
999 |
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta') |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1000 |
show "integral\<^isup>L ?P ?f \<le> 1" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1001 |
using neg le1 fin positive_integral_positive[of ?P ?f] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1002 |
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1003 |
qed |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1004 |
also have "- log b (integral\<^isup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1005 |
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
|
1006 |
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
|
1007 |
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
|
1008 |
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
|
1009 |
by eventually_elim (auto simp: divide_pos_pos mult_pos_pos) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1010 |
show "integrable ?P ?f" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1011 |
unfolding integrable_def |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1012 |
using fin neg by (auto intro!: M simp: split_beta') |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1013 |
show "integrable ?P (\<lambda>x. - log b (?f x))" |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1014 |
apply (subst integral_density) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1015 |
apply (auto intro!: M) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1016 |
apply (auto intro!: M distributed_real_AE[OF Pxyz]) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1017 |
apply (auto intro!: M borel_measurable_uminus borel_measurable_log simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1018 |
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3]) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1019 |
apply (auto intro!: M borel_measurable_uminus borel_measurable_log simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1020 |
apply (auto intro!: M borel_measurable_uminus borel_measurable_log simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1021 |
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
|
1022 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1023 |
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
|
1024 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1025 |
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
|
1026 |
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
|
1027 |
unfolding `?eq` |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1028 |
apply (subst integral_density) |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1029 |
apply (auto intro!: M) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1030 |
apply (auto intro!: M distributed_real_AE[OF Pxyz]) [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1031 |
apply (auto intro!: M borel_measurable_uminus borel_measurable_log simp: split_beta') [] |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1032 |
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
|
1033 |
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
|
1034 |
apply eventually_elim |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1035 |
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
|
1036 |
done |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1037 |
finally show ?nonneg |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1038 |
by simp |
40859 | 1039 |
qed |
1040 |
||
1041 |
lemma (in information_space) conditional_mutual_information_eq: |
|
47694 | 1042 |
assumes Pz: "simple_distributed M Z Pz" |
1043 |
assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz" |
|
1044 |
assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz" |
|
1045 |
assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
1046 |
shows "\<I>(X ; Y | Z) = |
|
1047 |
(\<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))))" |
|
1048 |
proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _ |
|
1049 |
simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz] |
|
1050 |
simple_distributed_joint2[OF Pxyz]]) |
|
1051 |
note simple_distributed_joint2_finite[OF Pxyz, simp] |
|
1052 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
1053 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1054 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
1055 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1056 |
show "sigma_finite_measure (count_space (Z ` space M))" |
|
1057 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1058 |
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) = |
|
1059 |
count_space (X`space M \<times> Y`space M \<times> Z`space M)" |
|
1060 |
(is "?P = ?C") |
|
1061 |
by (simp add: pair_measure_count_space) |
|
40859 | 1062 |
|
47694 | 1063 |
let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)" |
1064 |
have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^isub>M count_space (Z ` space M))" |
|
1065 |
using simple_distributed_joint[OF Pxz] by (rule distributed_measurable) |
|
1066 |
from measurable_comp[OF this measurable_fst] |
|
1067 |
have "random_variable (count_space (X ` space M)) X" |
|
1068 |
by (simp add: comp_def) |
|
1069 |
then have "simple_function M X" |
|
1070 |
unfolding simple_function_def by auto |
|
1071 |
then have "simple_distributed M X ?Px" |
|
1072 |
by (rule simple_distributedI) auto |
|
1073 |
then show "distributed M (count_space (X ` space M)) X ?Px" |
|
1074 |
by (rule simple_distributed) |
|
1075 |
||
1076 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)" |
|
1077 |
let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)" |
|
1078 |
let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)" |
|
1079 |
show |
|
1080 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))" |
|
1081 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))" |
|
1082 |
by (auto intro!: integrable_count_space simp: pair_measure_count_space) |
|
1083 |
let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))" |
|
1084 |
let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))" |
|
1085 |
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)" |
|
1086 |
by (auto intro!: ext) |
|
1087 |
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)" |
|
1088 |
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta') |
|
36624 | 1089 |
qed |
1090 |
||
47694 | 1091 |
lemma (in information_space) conditional_mutual_information_nonneg: |
1092 |
assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z" |
|
1093 |
shows "0 \<le> \<I>(X ; Y | Z)" |
|
1094 |
proof - |
|
49787
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1095 |
have [simp]: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) = |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1096 |
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
|
1097 |
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
|
1098 |
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
|
1099 |
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
|
1100 |
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
|
1101 |
show ?thesis |
d8de705b48d4
rule to show that conditional mutual information is non-negative in the continuous case
hoelzl
parents:
49786
diff
changeset
|
1102 |
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
|
1103 |
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
|
1104 |
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
|
1105 |
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
|
1106 |
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
|
1107 |
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
|
1108 |
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
|
1109 |
done |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1110 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1111 |
|
39097 | 1112 |
subsection {* Conditional Entropy *} |
1113 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1114 |
definition (in prob_space) |
47694 | 1115 |
"conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1116 |
|
40859 | 1117 |
abbreviation (in information_space) |
1118 |
conditional_entropy_Pow ("\<H>'(_ | _')") where |
|
47694 | 1119 |
"\<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
|
1120 |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1121 |
lemma (in information_space) |
47694 | 1122 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
1123 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1124 |
assumes Px: "distributed M S X Px" |
|
1125 |
assumes Py: "distributed M T Y Py" |
|
1126 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
1127 |
assumes I1: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
|
1128 |
assumes I2: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1129 |
shows conditional_entropy_generic_eq: "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^isub>M T))" (is ?eq) |
40859 | 1130 |
proof - |
47694 | 1131 |
interpret S: sigma_finite_measure S by fact |
1132 |
interpret T: sigma_finite_measure T by fact |
|
1133 |
interpret ST: pair_sigma_finite S T .. |
|
1134 |
have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" .. |
|
1135 |
||
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1136 |
let ?P = "density (S \<Otimes>\<^isub>M T) Pxy" |
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1137 |
interpret Pxy: prob_space ?P |
47694 | 1138 |
unfolding Pxy[THEN distributed_distr_eq_density, symmetric] |
1139 |
using Pxy[THEN distributed_measurable] by (rule prob_space_distr) |
|
1140 |
||
1141 |
from Py Pxy have distr_eq: "distr M T Y = |
|
1142 |
distr (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) T snd" |
|
1143 |
by (subst distr_distr[OF measurable_snd]) (auto dest: distributed_measurable simp: comp_def) |
|
1144 |
||
1145 |
have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)" |
|
49786 | 1146 |
by (rule entropy_distr[OF Py]) |
47694 | 1147 |
also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" |
1148 |
using b_gt_1 Py[THEN distributed_real_measurable] |
|
1149 |
by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong) |
|
1150 |
finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" . |
|
1151 |
||
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1152 |
have ae1: "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 1153 |
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair) |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1154 |
moreover have ae2: "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
47694 | 1155 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1156 |
moreover have ae3: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)" |
47694 | 1157 |
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1158 |
moreover have ae4: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)" |
47694 | 1159 |
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
|
1160 |
moreover note ae5 = Pxy[THEN distributed_real_AE] |
47694 | 1161 |
ultimately have pos: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Pxy x \<and> 0 \<le> Px (fst x) \<and> 0 \<le> Py (snd x) \<and> |
1162 |
(Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Px (fst x) \<and> 0 < Py (snd x)))" |
|
1163 |
by eventually_elim auto |
|
1164 |
||
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1165 |
from pos have ae: "AE x in S \<Otimes>\<^isub>M T. |
47694 | 1166 |
Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))" |
1167 |
by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1) |
|
49788
3c10763f5cb4
show and use distributed_swap and distributed_jointI
hoelzl
parents:
49787
diff
changeset
|
1168 |
with I1 I2 show ?eq |
40859 | 1169 |
unfolding conditional_entropy_def |
47694 | 1170 |
apply (subst e_eq) |
49786 | 1171 |
apply (subst entropy_distr[OF Pxy]) |
47694 | 1172 |
unfolding minus_diff_minus |
1173 |
apply (subst integral_diff(2)[symmetric]) |
|
1174 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
|
1175 |
done |
|
39097 | 1176 |
qed |
1177 |
||
40859 | 1178 |
lemma (in information_space) conditional_entropy_eq: |
47694 | 1179 |
assumes Y: "simple_distributed M Y Py" and X: "simple_function M X" |
1180 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
1181 |
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))" |
|
1182 |
proof (subst conditional_entropy_generic_eq[OF _ _ |
|
1183 |
simple_distributed[OF simple_distributedI[OF X refl]] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
|
1184 |
have [simp]: "finite (X`space M)" using X by (simp add: simple_function_def) |
|
1185 |
note Y[THEN simple_distributed_finite, simp] |
|
1186 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
1187 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1188 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
1189 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1190 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
1191 |
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)" |
|
1192 |
(is "?P = ?C") |
|
1193 |
using X Y by (simp add: simple_distributed_finite pair_measure_count_space) |
|
1194 |
with X Y show |
|
1195 |
"integrable ?P (\<lambda>x. ?f x * log b (?f x))" |
|
1196 |
"integrable ?P (\<lambda>x. ?f x * log b (Py (snd x)))" |
|
1197 |
by (auto intro!: integrable_count_space simp: simple_distributed_finite) |
|
1198 |
have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) = |
|
1199 |
(\<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)" |
|
1200 |
by auto |
|
1201 |
from X Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) = |
|
1202 |
- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))" |
|
1203 |
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta') |
|
1204 |
qed |
|
39097 | 1205 |
|
47694 | 1206 |
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
|
1207 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
47694 | 1208 |
shows "\<I>(X ; X | Y) = \<H>(X | Y)" |
1209 |
proof - |
|
1210 |
def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0" |
|
1211 |
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" |
|
1212 |
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" |
|
1213 |
let ?M = "X`space M \<times> X`space M \<times> Y`space M" |
|
39097 | 1214 |
|
47694 | 1215 |
note XY = simple_function_Pair[OF X Y] |
1216 |
note XXY = simple_function_Pair[OF X XY] |
|
1217 |
have Py: "simple_distributed M Y Py" |
|
1218 |
using Y by (rule simple_distributedI) (auto simp: Py_def) |
|
1219 |
have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
1220 |
using XY by (rule simple_distributedI) (auto simp: Pxy_def) |
|
1221 |
have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy" |
|
1222 |
using XXY by (rule simple_distributedI) (auto simp: Pxxy_def) |
|
1223 |
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" |
|
1224 |
by auto |
|
1225 |
have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A" |
|
1226 |
by (auto simp: inj_on_def) |
|
1227 |
have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)" |
|
1228 |
by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob]) |
|
1229 |
have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1230 |
by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair) |
|
1231 |
then show ?thesis |
|
1232 |
apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy]) |
|
1233 |
apply (subst conditional_entropy_eq[OF Py X Pxy]) |
|
1234 |
apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj] |
|
1235 |
log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space) |
|
1236 |
using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE] |
|
1237 |
apply (auto simp add: not_le[symmetric] AE_count_space) |
|
1238 |
done |
|
1239 |
qed |
|
1240 |
||
1241 |
lemma (in information_space) conditional_entropy_nonneg: |
|
1242 |
assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)" |
|
1243 |
using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y] |
|
1244 |
by simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1245 |
|
39097 | 1246 |
subsection {* Equalities *} |
1247 |
||
47694 | 1248 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr: |
1249 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real" |
|
1250 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1251 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
1252 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
1253 |
assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))" |
|
1254 |
assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
|
1255 |
assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
|
1256 |
shows "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
40859 | 1257 |
proof - |
47694 | 1258 |
have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^isub>M T))" |
1259 |
using b_gt_1 Px[THEN distributed_real_measurable] |
|
49786 | 1260 |
apply (subst entropy_distr[OF Px]) |
47694 | 1261 |
apply (subst distributed_transform_integral[OF Pxy Px, where T=fst]) |
1262 |
apply (auto intro!: integral_cong) |
|
1263 |
done |
|
1264 |
||
1265 |
have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))" |
|
1266 |
using b_gt_1 Py[THEN distributed_real_measurable] |
|
49786 | 1267 |
apply (subst entropy_distr[OF Py]) |
47694 | 1268 |
apply (subst distributed_transform_integral[OF Pxy Py, where T=snd]) |
1269 |
apply (auto intro!: integral_cong) |
|
1270 |
done |
|
1271 |
||
1272 |
interpret S: sigma_finite_measure S by fact |
|
1273 |
interpret T: sigma_finite_measure T by fact |
|
1274 |
interpret ST: pair_sigma_finite S T .. |
|
1275 |
have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" .. |
|
1276 |
||
1277 |
have XY: "entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T))" |
|
49786 | 1278 |
by (subst entropy_distr[OF Pxy]) (auto intro!: integral_cong) |
47694 | 1279 |
|
1280 |
have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
|
1281 |
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair) |
|
1282 |
moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1283 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
|
1284 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)" |
|
1285 |
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
|
1286 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)" |
|
1287 |
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
|
1288 |
moreover note Pxy[THEN distributed_real_AE] |
|
1289 |
ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) = |
|
1290 |
Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
|
1291 |
(is "AE x in _. ?f x = ?g x") |
|
1292 |
proof eventually_elim |
|
1293 |
case (goal1 x) |
|
1294 |
show ?case |
|
1295 |
proof cases |
|
1296 |
assume "Pxy x \<noteq> 0" |
|
1297 |
with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x" |
|
1298 |
by auto |
|
1299 |
then show ?thesis |
|
1300 |
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps) |
|
1301 |
qed simp |
|
1302 |
qed |
|
1303 |
||
1304 |
have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?f" |
|
1305 |
unfolding X Y XY |
|
1306 |
apply (subst integral_diff) |
|
1307 |
apply (intro integral_diff Ixy Ix Iy)+ |
|
1308 |
apply (subst integral_diff) |
|
1309 |
apply (intro integral_diff Ixy Ix Iy)+ |
|
1310 |
apply (simp add: field_simps) |
|
1311 |
done |
|
1312 |
also have "\<dots> = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?g" |
|
1313 |
using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE) |
|
1314 |
also have "\<dots> = mutual_information b S T X Y" |
|
1315 |
by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric]) |
|
1316 |
finally show ?thesis .. |
|
1317 |
qed |
|
1318 |
||
1319 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy: |
|
1320 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1321 |
shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)" |
|
1322 |
proof - |
|
1323 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1324 |
using sf_X by (rule simple_distributedI) auto |
|
1325 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1326 |
using sf_Y by (rule simple_distributedI) auto |
|
1327 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1328 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1329 |
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))" |
|
1330 |
by (rule simple_distributedI) auto |
|
1331 |
from simple_distributed_joint_finite[OF this, simp] |
|
1332 |
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
|
1333 |
by (simp add: pair_measure_count_space) |
|
1334 |
||
1335 |
have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))" |
|
1336 |
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] |
|
1337 |
by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space) |
|
1338 |
then show ?thesis |
|
1339 |
unfolding conditional_entropy_def by simp |
|
1340 |
qed |
|
1341 |
||
1342 |
lemma (in information_space) mutual_information_nonneg_simple: |
|
1343 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1344 |
shows "0 \<le> \<I>(X ; Y)" |
|
1345 |
proof - |
|
1346 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1347 |
using sf_X by (rule simple_distributedI) auto |
|
1348 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1349 |
using sf_Y by (rule simple_distributedI) auto |
|
1350 |
||
1351 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1352 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1353 |
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))" |
|
1354 |
by (rule simple_distributedI) auto |
|
1355 |
||
1356 |
from simple_distributed_joint_finite[OF this, simp] |
|
1357 |
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
|
1358 |
by (simp add: pair_measure_count_space) |
|
1359 |
||
40859 | 1360 |
show ?thesis |
47694 | 1361 |
by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
1362 |
(simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite) |
|
40859 | 1363 |
qed |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1364 |
|
40859 | 1365 |
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
|
1366 |
assumes X: "simple_function M X" and Z: "simple_function M Z" |
40859 | 1367 |
shows "\<H>(X | Z) \<le> \<H>(X)" |
36624 | 1368 |
proof - |
47694 | 1369 |
have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple) |
1370 |
also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] . |
|
1371 |
finally show ?thesis by auto |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1372 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1373 |
|
40859 | 1374 |
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
|
1375 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
40859 | 1376 |
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)" |
1377 |
proof - |
|
47694 | 1378 |
note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl] |
1379 |
note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl] |
|
1380 |
note simple_distributed_joint_finite[OF this, simp] |
|
1381 |
let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" |
|
1382 |
let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)" |
|
1383 |
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" |
|
1384 |
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))" |
|
1385 |
using XY by (rule entropy_simple_distributed) |
|
1386 |
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))" |
|
1387 |
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)"]) |
|
1388 |
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))" |
|
1389 |
by (auto intro!: setsum_cong) |
|
1390 |
also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" |
|
49786 | 1391 |
by (subst entropy_distr[OF simple_distributed_joint[OF YX]]) |
47694 | 1392 |
(auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite |
1393 |
cong del: setsum_cong intro!: setsum_mono_zero_left) |
|
1394 |
finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" . |
|
1395 |
then show ?thesis |
|
1396 |
unfolding conditional_entropy_def by simp |
|
36624 | 1397 |
qed |
1398 |
||
40859 | 1399 |
lemma (in information_space) entropy_partition: |
47694 | 1400 |
assumes X: "simple_function M X" |
1401 |
shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)" |
|
36624 | 1402 |
proof - |
47694 | 1403 |
note fX = simple_function_compose[OF X, of f] |
1404 |
have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto |
|
1405 |
have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A" |
|
1406 |
by (auto simp: inj_on_def) |
|
1407 |
show ?thesis |
|
1408 |
apply (subst entropy_chain_rule[symmetric, OF fX X]) |
|
1409 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]]) |
|
1410 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1411 |
unfolding eq |
|
1412 |
apply (subst setsum_reindex[OF inj]) |
|
1413 |
apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"]) |
|
1414 |
done |
|
36624 | 1415 |
qed |
1416 |
||
40859 | 1417 |
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
|
1418 |
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)" |
40859 | 1419 |
proof - |
47694 | 1420 |
note fX = simple_function_compose[OF X, of f] |
1421 |
from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition) |
|
40859 | 1422 |
then show "\<H>(f \<circ> X) \<le> \<H>(X)" |
47694 | 1423 |
by (auto intro: conditional_entropy_nonneg[OF X fX]) |
40859 | 1424 |
qed |
36624 | 1425 |
|
40859 | 1426 |
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
|
1427 |
assumes X: "simple_function M X" and inj: "inj_on f (X`space M)" |
36624 | 1428 |
shows "\<H>(f \<circ> X) = \<H>(X)" |
1429 |
proof (rule antisym) |
|
40859 | 1430 |
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] . |
36624 | 1431 |
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
|
1432 |
have sf: "simple_function M (f \<circ> X)" |
40859 | 1433 |
using X by auto |
36624 | 1434 |
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))" |
47694 | 1435 |
unfolding o_assoc |
1436 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1437 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"]) |
|
1438 |
apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def) |
|
1439 |
done |
|
36624 | 1440 |
also have "... \<le> \<H>(f \<circ> X)" |
40859 | 1441 |
using entropy_data_processing[OF sf] . |
36624 | 1442 |
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" . |
1443 |
qed |
|
1444 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1445 |
end |