author | hoelzl |
Wed, 10 Oct 2012 12:12:18 +0200 | |
changeset 49776 | 199d1d5bb17e |
parent 47694 | 05663f75964c |
child 49785 | 0a8adca22974 |
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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theory Information |
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imports |
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Independent_Family |
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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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section "Convex theory" |
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lemma log_setsum: |
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assumes "finite s" "s \<noteq> {}" |
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assumes "b > 1" |
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assumes "(\<Sum> i \<in> s. a i) = 1" |
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assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0" |
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assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}" |
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shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))" |
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proof - |
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have "convex_on {0 <..} (\<lambda> x. - log b x)" |
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by (rule minus_log_convex[OF `b > 1`]) |
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hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))" |
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using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastforce |
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thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le) |
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qed |
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lemma log_setsum': |
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assumes "finite s" "s \<noteq> {}" |
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assumes "b > 1" |
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assumes "(\<Sum> i \<in> s. a i) = 1" |
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assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i" |
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"\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i" |
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shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))" |
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proof - |
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have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)" |
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using assms by (auto intro!: setsum_mono_zero_cong_left) |
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moreover have "log b (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) \<ge> (\<Sum> i \<in> s - {i. a i = 0}. a i * log b (y i))" |
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proof (rule log_setsum) |
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have "setsum a (s - {i. a i = 0}) = setsum a s" |
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using assms(1) by (rule setsum_mono_zero_cong_left) auto |
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thus sum_1: "setsum a (s - {i. a i = 0}) = 1" |
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"finite (s - {i. a i = 0})" using assms by simp_all |
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show "s - {i. a i = 0} \<noteq> {}" |
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proof |
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assume *: "s - {i. a i = 0} = {}" |
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hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty) |
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with sum_1 show False by simp |
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qed |
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fix i assume "i \<in> s - {i. a i = 0}" |
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hence "i \<in> s" "a i \<noteq> 0" by simp_all |
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thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto |
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qed fact+ |
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ultimately show ?thesis by simp |
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qed |
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lemma log_setsum_divide: |
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assumes "finite S" and "S \<noteq> {}" and "1 < b" |
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assumes "(\<Sum>x\<in>S. g x) = 1" |
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assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0" |
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assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x" |
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shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)" |
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proof - |
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have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y" |
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using `1 < b` by (subst log_le_cancel_iff) auto |
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have "- (\<Sum>x\<in>S. g x * log b (g x / f x)) = (\<Sum>x\<in>S. g x * log b (f x / g x))" |
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proof (unfold setsum_negf[symmetric], rule setsum_cong) |
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fix x assume x: "x \<in> S" |
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show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)" |
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proof (cases "g x = 0") |
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case False |
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with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all |
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thus ?thesis using `1 < b` by (simp add: log_divide field_simps) |
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qed simp |
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qed rule |
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also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))" |
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proof (rule log_setsum') |
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fix x assume x: "x \<in> S" "0 < g x" |
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with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos) |
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qed fact+ |
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also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S` |
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by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"] |
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split: split_if_asm) |
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also have "... \<le> log b (\<Sum>x\<in>S. f x)" |
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proof (rule log_mono) |
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have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp |
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also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum") |
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proof (rule setsum_strict_mono) |
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show "finite (S - {x. g x = 0})" using `finite S` by simp |
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show "S - {x. g x = 0} \<noteq> {}" |
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proof |
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assume "S - {x. g x = 0} = {}" |
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hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto |
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with `(\<Sum>x\<in>S. g x) = 1` show False by simp |
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qed |
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fix x assume "x \<in> S - {x. g x = 0}" |
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thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto |
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qed |
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finally show "0 < ?sum" . |
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show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)" |
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using `finite S` pos by (auto intro!: setsum_mono2) |
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qed |
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finally show ?thesis . |
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qed |
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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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47694 | 233 |
lemma (in information_space) KL_gt_0: |
234 |
fixes D :: "'a \<Rightarrow> real" |
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assumes "prob_space (density M D)" |
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assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
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assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
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assumes A: "density M D \<noteq> M" |
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shows "0 < KL_divergence b M (density M D)" |
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proof - |
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interpret N: prob_space "density M D" by fact |
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47694 | 243 |
obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A" |
244 |
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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47694 | 253 |
have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)" |
254 |
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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257 |
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47694 | 258 |
have "integrable M D" "integral\<^isup>L M D = 1" |
259 |
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) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
263 |
also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
264 |
using `integrable M D` `integral\<^isup>L M D = 1` |
47694 | 265 |
by (simp add: emeasure_eq_measure) |
266 |
also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
267 |
proof (rule integral_less_AE) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
268 |
show "integrable M (\<lambda>x. D x - indicator ?D_set x)" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
269 |
using `integrable M D` |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
270 |
by (intro integral_diff integral_indicator) auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
271 |
next |
47694 | 272 |
from integral_cmult(1)[OF int, of "ln b"] |
273 |
show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" |
|
274 |
by (simp add: ac_simps) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
275 |
next |
47694 | 276 |
show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
277 |
proof |
47694 | 278 |
assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0" |
279 |
then have disj: "AE x in M. D x = 1 \<or> D x = 0" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
280 |
using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
281 |
|
47694 | 282 |
have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
283 |
using D(1) by auto |
47694 | 284 |
also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) \<partial>M)" |
43920 | 285 |
using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def) |
47694 | 286 |
finally have "AE x in M. D x = 1" |
287 |
using D D_pos by (intro AE_I_eq_1) auto |
|
43920 | 288 |
then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)" |
289 |
by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric]) |
|
47694 | 290 |
also have "\<dots> = density M D A" |
291 |
using `A \<in> sets M` D by (simp add: emeasure_density) |
|
292 |
finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
293 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
294 |
show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M" |
47694 | 295 |
using D(1) by (auto intro: sets_Collect_conj) |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
296 |
|
47694 | 297 |
show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow> |
298 |
D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
299 |
using D(2) |
47694 | 300 |
proof (eventually_elim, safe) |
301 |
fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t" |
|
302 |
and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
303 |
|
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
304 |
have "D t - 1 = D t - indicator ?D_set t" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
305 |
using Dt by simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
306 |
also note eq |
47694 | 307 |
also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)" |
308 |
using b_gt_1 `D t \<noteq> 0` `0 \<le> D t` |
|
309 |
by (simp add: log_def ln_div less_le) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
310 |
finally have "ln (1 / D t) = 1 / D t - 1" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
311 |
using `D t \<noteq> 0` by (auto simp: field_simps) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
312 |
from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1` |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
313 |
show False by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
314 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
315 |
|
47694 | 316 |
show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
317 |
using D(2) AE_space |
|
318 |
proof eventually_elim |
|
319 |
fix t assume "t \<in> space M" "0 \<le> D t" |
|
320 |
show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
321 |
proof cases |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
322 |
assume asm: "D t \<noteq> 0" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
323 |
then have "0 < D t" using `0 \<le> D t` by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
324 |
then have "0 < 1 / D t" by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
325 |
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
|
326 |
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
|
327 |
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
|
328 |
using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto |
47694 | 329 |
also have "\<dots> = D t * (ln b * log b (D t))" |
330 |
using `0 < D t` b_gt_1 |
|
331 |
by (simp_all add: log_def ln_div) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
332 |
finally show ?thesis by simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
333 |
qed simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
334 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
335 |
qed |
47694 | 336 |
also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)" |
337 |
by (simp add: ac_simps) |
|
338 |
also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)" |
|
339 |
using int by (rule integral_cmult) |
|
340 |
finally show ?thesis |
|
341 |
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
|
342 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
343 |
|
47694 | 344 |
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
|
345 |
proof - |
47694 | 346 |
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
|
347 |
proof (rule RN_deriv_unique) |
47694 | 348 |
show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto |
349 |
show "density M (\<lambda>x. 1) = M" |
|
350 |
apply (auto intro!: measure_eqI emeasure_density) |
|
351 |
apply (subst emeasure_density) |
|
352 |
apply auto |
|
353 |
done |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
354 |
qed |
47694 | 355 |
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
|
356 |
by (elim AE_mp) simp |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
357 |
from integral_cong_AE[OF this] |
47694 | 358 |
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
|
359 |
by (simp add: entropy_density_def comp_def) |
47694 | 360 |
then show "KL_divergence b M M = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
361 |
unfolding KL_divergence_def |
47694 | 362 |
by auto |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
363 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
364 |
|
47694 | 365 |
lemma (in information_space) KL_eq_0_iff_eq: |
366 |
fixes D :: "'a \<Rightarrow> real" |
|
367 |
assumes "prob_space (density M D)" |
|
368 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
369 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
370 |
shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M" |
|
371 |
using KL_same_eq_0[of b] KL_gt_0[OF assms] |
|
372 |
by (auto simp: less_le) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
373 |
|
47694 | 374 |
lemma (in information_space) KL_eq_0_iff_eq_ac: |
375 |
fixes D :: "'a \<Rightarrow> real" |
|
376 |
assumes "prob_space N" |
|
377 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
|
378 |
assumes int: "integrable N (entropy_density b M N)" |
|
379 |
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
|
380 |
proof - |
47694 | 381 |
interpret N: prob_space N by fact |
382 |
have "finite_measure N" by unfold_locales |
|
383 |
from real_RN_deriv[OF this ac] guess D . note D = this |
|
384 |
||
385 |
have "N = density M (RN_deriv M N)" |
|
386 |
using ac by (rule density_RN_deriv[symmetric]) |
|
387 |
also have "\<dots> = density M D" |
|
388 |
using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong) |
|
389 |
finally have N: "N = density M D" . |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
390 |
|
47694 | 391 |
from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density |
392 |
have "integrable N (\<lambda>x. log b (D x))" |
|
393 |
by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int]) |
|
394 |
(auto simp: N entropy_density_def) |
|
395 |
with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))" |
|
396 |
by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def) |
|
397 |
with `prob_space N` D show ?thesis |
|
398 |
unfolding N |
|
399 |
by (intro KL_eq_0_iff_eq) auto |
|
41833
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
400 |
qed |
563bea92b2c0
add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents:
41689
diff
changeset
|
401 |
|
47694 | 402 |
lemma (in information_space) KL_nonneg: |
403 |
assumes "prob_space (density M D)" |
|
404 |
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x" |
|
405 |
assumes int: "integrable M (\<lambda>x. D x * log b (D x))" |
|
406 |
shows "0 \<le> KL_divergence b M (density M D)" |
|
407 |
using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0) |
|
40859 | 408 |
|
47694 | 409 |
lemma (in sigma_finite_measure) KL_density_density_nonneg: |
410 |
fixes f g :: "'a \<Rightarrow> real" |
|
411 |
assumes "1 < b" |
|
412 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)" |
|
413 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)" |
|
414 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
415 |
assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))" |
|
416 |
shows "0 \<le> KL_divergence b (density M f) (density M g)" |
|
417 |
proof - |
|
418 |
interpret Mf: prob_space "density M f" by fact |
|
419 |
interpret Mf: information_space "density M f" b by default fact |
|
420 |
have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _") |
|
421 |
using f g ac by (subst density_density_divide) simp_all |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
422 |
|
47694 | 423 |
have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))" |
424 |
proof (rule Mf.KL_nonneg) |
|
425 |
show "prob_space ?DD" unfolding eq by fact |
|
426 |
from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)" |
|
427 |
by auto |
|
428 |
show "AE x in density M f. 0 \<le> g x / f x" |
|
429 |
using f g by (auto simp: AE_density divide_nonneg_nonneg) |
|
430 |
show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))" |
|
431 |
using `1 < b` f g ac |
|
432 |
by (subst integral_density) |
|
433 |
(auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If) |
|
434 |
qed |
|
435 |
also have "\<dots> = KL_divergence b (density M f) (density M g)" |
|
436 |
using f g ac by (subst density_density_divide) simp_all |
|
437 |
finally show ?thesis . |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
438 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
439 |
|
39097 | 440 |
subsection {* Mutual Information *} |
441 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
442 |
definition (in prob_space) |
38656 | 443 |
"mutual_information b S T X Y = |
47694 | 444 |
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
|
445 |
|
47694 | 446 |
lemma (in information_space) mutual_information_indep_vars: |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
447 |
fixes S T X Y |
47694 | 448 |
defines "P \<equiv> distr M S X \<Otimes>\<^isub>M distr M T Y" |
449 |
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
|
450 |
shows "indep_var S X T Y \<longleftrightarrow> |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
451 |
(random_variable S X \<and> random_variable T Y \<and> |
47694 | 452 |
absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and> |
453 |
mutual_information b S T X Y = 0)" |
|
454 |
unfolding indep_var_distribution_eq |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
455 |
proof safe |
47694 | 456 |
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
|
457 |
|
47694 | 458 |
interpret X: prob_space "distr M S X" |
459 |
by (rule prob_space_distr) fact |
|
460 |
interpret Y: prob_space "distr M T Y" |
|
461 |
by (rule prob_space_distr) fact |
|
462 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default |
|
463 |
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
|
464 |
|
47694 | 465 |
interpret Q: prob_space Q unfolding Q_def |
466 |
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
|
467 |
|
47694 | 468 |
{ 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))" |
469 |
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
|
470 |
|
47694 | 471 |
show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def) |
472 |
then have ed: "entropy_density b P Q \<in> borel_measurable P" |
|
473 |
by (rule P.measurable_entropy_density) simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
474 |
|
47694 | 475 |
have "AE x in P. 1 = RN_deriv P Q x" |
476 |
proof (rule P.RN_deriv_unique) |
|
477 |
show "density P (\<lambda>x. 1) = Q" |
|
478 |
unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density) |
|
479 |
qed auto |
|
480 |
then have ae_0: "AE x in P. entropy_density b P Q x = 0" |
|
481 |
by eventually_elim (auto simp: entropy_density_def) |
|
482 |
then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)" |
|
483 |
using ed unfolding `Q = P` by (intro integrable_cong_AE) auto |
|
484 |
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
|
485 |
|
47694 | 486 |
show "mutual_information b S T X Y = 0" |
487 |
unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P` |
|
488 |
using ae_0 by (simp cong: integral_cong_AE) } |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
489 |
|
47694 | 490 |
{ assume ac: "absolutely_continuous P Q" |
491 |
assume int: "integrable Q (entropy_density b P Q)" |
|
492 |
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
|
493 |
|
47694 | 494 |
have eq: "Q = P" |
495 |
proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1]) |
|
496 |
show "prob_space Q" by unfold_locales |
|
497 |
show "absolutely_continuous P Q" by fact |
|
498 |
show "integrable Q (entropy_density b P Q)" by fact |
|
499 |
show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure) |
|
500 |
show "KL_divergence b P Q = 0" |
|
501 |
using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def) |
|
502 |
qed |
|
503 |
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))" |
|
504 |
unfolding P_def Q_def .. } |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
505 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42148
diff
changeset
|
506 |
|
40859 | 507 |
abbreviation (in information_space) |
508 |
mutual_information_Pow ("\<I>'(_ ; _')") where |
|
47694 | 509 |
"\<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
|
510 |
|
47694 | 511 |
lemma (in information_space) |
512 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
513 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
|
514 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
515 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
516 |
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
|
517 |
shows mutual_information_distr: "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R") |
|
518 |
and mutual_information_nonneg: "integrable (S \<Otimes>\<^isub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y" |
|
40859 | 519 |
proof - |
47694 | 520 |
have X: "random_variable S X" |
521 |
using Px by (auto simp: distributed_def) |
|
522 |
have Y: "random_variable T Y" |
|
523 |
using Py by (auto simp: distributed_def) |
|
524 |
interpret S: sigma_finite_measure S by fact |
|
525 |
interpret T: sigma_finite_measure T by fact |
|
526 |
interpret ST: pair_sigma_finite S T .. |
|
527 |
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr) |
|
528 |
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr) |
|
529 |
interpret XY: pair_prob_space "distr M S X" "distr M T Y" .. |
|
530 |
let ?P = "S \<Otimes>\<^isub>M T" |
|
531 |
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))" |
|
532 |
||
533 |
{ fix A assume "A \<in> sets S" |
|
534 |
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)" |
|
535 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
536 |
intro!: arg_cong[where f="emeasure M"]) } |
|
537 |
note marginal_eq1 = this |
|
538 |
{ fix A assume "A \<in> sets T" |
|
539 |
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)" |
|
540 |
by (auto simp: emeasure_distr measurable_Pair measurable_space |
|
541 |
intro!: arg_cong[where f="emeasure M"]) } |
|
542 |
note marginal_eq2 = this |
|
543 |
||
544 |
have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))" |
|
545 |
by auto |
|
546 |
||
547 |
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)))" |
|
548 |
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq |
|
549 |
proof (subst pair_measure_density) |
|
550 |
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T" |
|
551 |
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)" |
|
552 |
using Px Py by (auto simp: distributed_def) |
|
553 |
show "sigma_finite_measure (density S Px)" unfolding Px(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
554 |
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] .. |
|
555 |
qed (fact | simp)+ |
|
556 |
||
557 |
have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))" |
|
558 |
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] .. |
|
559 |
||
560 |
from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P" |
|
561 |
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'') |
|
562 |
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)" |
|
563 |
proof (rule ST.AE_pair_measure) |
|
564 |
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P" |
|
565 |
using f by auto |
|
566 |
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))" |
|
567 |
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE) |
|
568 |
qed |
|
569 |
||
570 |
have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)" |
|
571 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
572 |
moreover |
|
573 |
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)" |
|
574 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
575 |
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
576 |
by eventually_elim auto |
|
577 |
||
578 |
show "?M = ?R" |
|
579 |
unfolding M f_def |
|
580 |
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac |
|
581 |
by (rule ST.KL_density_density) |
|
582 |
||
583 |
assume int: "integrable (S \<Otimes>\<^isub>M T) f" |
|
584 |
show "0 \<le> ?M" unfolding M |
|
585 |
proof (rule ST.KL_density_density_nonneg |
|
586 |
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]]) |
|
587 |
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) " |
|
588 |
unfolding distributed_distr_eq_density[OF Pxy, symmetric] |
|
589 |
using distributed_measurable[OF Pxy] by (rule prob_space_distr) |
|
590 |
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))" |
|
591 |
unfolding distr_eq[symmetric] by unfold_locales |
|
40859 | 592 |
qed |
593 |
qed |
|
594 |
||
595 |
lemma (in information_space) |
|
47694 | 596 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
597 |
assumes "sigma_finite_measure S" "sigma_finite_measure T" |
|
598 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
599 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
600 |
assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y" |
|
601 |
shows mutual_information_eq_0: "mutual_information b S T X Y = 0" |
|
36624 | 602 |
proof - |
47694 | 603 |
interpret S: sigma_finite_measure S by fact |
604 |
interpret T: sigma_finite_measure T by fact |
|
605 |
interpret ST: pair_sigma_finite S T .. |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
606 |
|
47694 | 607 |
have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
608 |
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto |
|
609 |
moreover |
|
610 |
have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
611 |
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto |
|
612 |
moreover |
|
613 |
have "AE x in S \<Otimes>\<^isub>M T. Pxy x = Px (fst x) * Py (snd x)" |
|
614 |
using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy] |
|
615 |
by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'') |
|
616 |
ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0" |
|
617 |
by eventually_elim simp |
|
618 |
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))" |
|
619 |
by (rule integral_cong_AE) |
|
620 |
then show ?thesis |
|
621 |
by (subst mutual_information_distr[OF assms(1-5)]) simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
622 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
623 |
|
47694 | 624 |
lemma (in information_space) mutual_information_simple_distributed: |
625 |
assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py" |
|
626 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
627 |
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)))" |
|
628 |
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
|
629 |
note fin = simple_distributed_joint_finite[OF XY, simp] |
|
630 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
631 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
632 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
633 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
634 |
let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
635 |
let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))" |
|
636 |
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)" |
|
637 |
by auto |
|
638 |
with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M))) = |
|
639 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))" |
|
640 |
by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta' |
|
641 |
intro!: setsum_cong) |
|
642 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
643 |
|
47694 | 644 |
lemma (in information_space) |
645 |
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
|
646 |
assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py" |
|
647 |
assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
648 |
assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)" |
|
649 |
shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0" |
|
650 |
proof (subst mutual_information_simple_distributed[OF Px Py Pxy]) |
|
651 |
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))) = |
|
652 |
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)" |
|
653 |
by (intro setsum_cong) (auto simp: ae) |
|
654 |
then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. |
|
655 |
Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp |
|
656 |
qed |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
657 |
|
39097 | 658 |
subsection {* Entropy *} |
659 |
||
47694 | 660 |
definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where |
661 |
"entropy b S X = - KL_divergence b S (distr M S X)" |
|
662 |
||
40859 | 663 |
abbreviation (in information_space) |
664 |
entropy_Pow ("\<H>'(_')") where |
|
47694 | 665 |
"\<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
|
666 |
|
47694 | 667 |
lemma (in information_space) entropy_distr: |
668 |
fixes X :: "'a \<Rightarrow> 'b" |
|
669 |
assumes "sigma_finite_measure MX" and X: "distributed M MX X f" |
|
670 |
shows "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" |
|
671 |
proof - |
|
672 |
interpret MX: sigma_finite_measure MX by fact |
|
673 |
from X show ?thesis |
|
674 |
unfolding entropy_def X[THEN distributed_distr_eq_density] |
|
675 |
by (subst MX.KL_density[OF b_gt_1]) (simp_all add: distributed_real_AE distributed_real_measurable) |
|
39097 | 676 |
qed |
36624 | 677 |
|
47694 | 678 |
lemma (in information_space) entropy_uniform: |
679 |
assumes "sigma_finite_measure MX" |
|
680 |
assumes A: "A \<in> sets MX" "emeasure MX A \<noteq> 0" "emeasure MX A \<noteq> \<infinity>" |
|
681 |
assumes X: "distributed M MX X (\<lambda>x. 1 / measure MX A * indicator A x)" |
|
682 |
shows "entropy b MX X = log b (measure MX A)" |
|
683 |
proof (subst entropy_distr[OF _ X]) |
|
684 |
let ?f = "\<lambda>x. 1 / measure MX A * indicator A x" |
|
685 |
have "- (\<integral>x. ?f x * log b (?f x) \<partial>MX) = |
|
686 |
- (\<integral>x. (log b (1 / measure MX A) / measure MX A) * indicator A x \<partial>MX)" |
|
687 |
by (auto intro!: integral_cong simp: indicator_def) |
|
688 |
also have "\<dots> = - log b (inverse (measure MX A))" |
|
689 |
using A by (subst integral_cmult(2)) |
|
690 |
(simp_all add: measure_def real_of_ereal_eq_0 integral_cmult inverse_eq_divide) |
|
691 |
also have "\<dots> = log b (measure MX A)" |
|
692 |
using b_gt_1 A by (subst log_inverse) (auto simp add: measure_def less_le real_of_ereal_eq_0 |
|
693 |
emeasure_nonneg real_of_ereal_pos) |
|
694 |
finally show "- (\<integral>x. ?f x * log b (?f x) \<partial>MX) = log b (measure MX A)" by simp |
|
695 |
qed fact+ |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
696 |
|
47694 | 697 |
lemma (in information_space) entropy_simple_distributed: |
698 |
fixes X :: "'a \<Rightarrow> 'b" |
|
699 |
assumes X: "simple_distributed M X f" |
|
700 |
shows "\<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))" |
|
701 |
proof (subst entropy_distr[OF _ simple_distributed[OF X]]) |
|
702 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
703 |
using X by (simp add: sigma_finite_measure_count_space_finite simple_distributed_def) |
|
704 |
show "- (\<integral>x. f x * log b (f x) \<partial>(count_space (X`space M))) = - (\<Sum>x\<in>X ` space M. f x * log b (f x))" |
|
705 |
using X by (auto simp add: lebesgue_integral_count_space_finite) |
|
39097 | 706 |
qed |
707 |
||
40859 | 708 |
lemma (in information_space) entropy_le_card_not_0: |
47694 | 709 |
assumes X: "simple_distributed M X f" |
710 |
shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))" |
|
39097 | 711 |
proof - |
47694 | 712 |
have "\<H>(X) = (\<Sum>x\<in>X`space M. f x * log b (1 / f x))" |
713 |
unfolding entropy_simple_distributed[OF X] setsum_negf[symmetric] |
|
714 |
using X by (auto dest: simple_distributed_nonneg intro!: setsum_cong simp: log_simps less_le) |
|
715 |
also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. f x * (1 / f x))" |
|
716 |
using not_empty b_gt_1 `simple_distributed M X f` |
|
717 |
by (intro log_setsum') (auto simp: simple_distributed_nonneg simple_distributed_setsum_space) |
|
718 |
also have "\<dots> = log b (\<Sum>x\<in>X`space M. if f x \<noteq> 0 then 1 else 0)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41833
diff
changeset
|
719 |
by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto |
39097 | 720 |
finally show ?thesis |
47694 | 721 |
using `simple_distributed M X f` by (auto simp: setsum_cases real_eq_of_nat) |
39097 | 722 |
qed |
723 |
||
40859 | 724 |
lemma (in information_space) entropy_le_card: |
47694 | 725 |
assumes "simple_distributed M X f" |
40859 | 726 |
shows "\<H>(X) \<le> log b (real (card (X ` space M)))" |
39097 | 727 |
proof cases |
47694 | 728 |
assume "X ` space M \<inter> {x. f x \<noteq> 0} = {}" |
729 |
then have "\<And>x. x\<in>X`space M \<Longrightarrow> f x = 0" by auto |
|
39097 | 730 |
moreover |
731 |
have "0 < card (X`space M)" |
|
47694 | 732 |
using `simple_distributed M X f` not_empty by (auto simp: card_gt_0_iff) |
39097 | 733 |
then have "log b 1 \<le> log b (real (card (X`space M)))" |
734 |
using b_gt_1 by (intro log_le) auto |
|
47694 | 735 |
ultimately show ?thesis using assms by (simp add: entropy_simple_distributed) |
39097 | 736 |
next |
47694 | 737 |
assume False: "X ` space M \<inter> {x. f x \<noteq> 0} \<noteq> {}" |
738 |
have "card (X ` space M \<inter> {x. f x \<noteq> 0}) \<le> card (X ` space M)" |
|
739 |
(is "?A \<le> ?B") using assms not_empty |
|
740 |
by (auto intro!: card_mono simp: simple_function_def simple_distributed_def) |
|
40859 | 741 |
note entropy_le_card_not_0[OF assms] |
39097 | 742 |
also have "log b (real ?A) \<le> log b (real ?B)" |
40859 | 743 |
using b_gt_1 False not_empty `?A \<le> ?B` assms |
47694 | 744 |
by (auto intro!: log_le simp: card_gt_0_iff simp: simple_distributed_def) |
39097 | 745 |
finally show ?thesis . |
746 |
qed |
|
747 |
||
748 |
subsection {* Conditional Mutual Information *} |
|
749 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
750 |
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
|
751 |
"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
|
752 |
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
|
753 |
mutual_information b MX MZ X Z" |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
754 |
|
40859 | 755 |
abbreviation (in information_space) |
756 |
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where |
|
36624 | 757 |
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b |
47694 | 758 |
(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
|
759 |
|
40859 | 760 |
lemma (in information_space) conditional_mutual_information_generic_eq: |
47694 | 761 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P" |
762 |
assumes Px: "distributed M S X Px" |
|
763 |
assumes Pz: "distributed M P Z Pz" |
|
764 |
assumes Pyz: "distributed M (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz" |
|
765 |
assumes Pxz: "distributed M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz" |
|
766 |
assumes Pxyz: "distributed M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
767 |
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))))" |
|
768 |
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)))" |
|
769 |
shows "conditional_mutual_information b S T P X Y Z |
|
770 |
= (\<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))" |
|
40859 | 771 |
proof - |
47694 | 772 |
interpret S: sigma_finite_measure S by fact |
773 |
interpret T: sigma_finite_measure T by fact |
|
774 |
interpret P: sigma_finite_measure P by fact |
|
775 |
interpret TP: pair_sigma_finite T P .. |
|
776 |
interpret SP: pair_sigma_finite S P .. |
|
777 |
interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T .. |
|
778 |
interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" .. |
|
779 |
have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" .. |
|
780 |
have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" .. |
|
781 |
have YZ: "random_variable (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x))" |
|
782 |
using Pyz by (simp add: distributed_measurable) |
|
783 |
||
784 |
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" |
|
785 |
using measurable_comp[OF _ Pxyz[THEN distributed_real_measurable]] by (auto simp: comp_def) |
|
786 |
||
787 |
{ fix f g h M |
|
788 |
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)" |
|
789 |
from measurable_comp[OF h Pxz[THEN distributed_real_measurable]] |
|
790 |
measurable_comp[OF f Px[THEN distributed_real_measurable]] |
|
791 |
measurable_comp[OF g Pz[THEN distributed_real_measurable]] |
|
792 |
have "(\<lambda>x. log b (Pxz (h x) / (Px (f x) * Pz (g x)))) \<in> borel_measurable M" |
|
793 |
by (simp add: comp_def b_gt_1) } |
|
794 |
note borel_log = this |
|
795 |
||
796 |
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)" |
|
797 |
by (auto simp add: split_beta' comp_def intro!: measurable_Pair measurable_snd') |
|
798 |
||
799 |
from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) = |
|
800 |
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))" |
|
801 |
by (subst distr_distr[OF measurable_cut]) (auto dest: distributed_measurable simp: comp_def) |
|
40859 | 802 |
|
47694 | 803 |
have "mutual_information b S P X Z = |
804 |
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))" |
|
805 |
by (rule mutual_information_distr[OF S P Px Pz Pxz]) |
|
806 |
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))" |
|
807 |
using b_gt_1 Pxz Px Pz |
|
808 |
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) |
|
809 |
(auto simp: split_beta' intro!: measurable_Pair measurable_snd' measurable_snd'' measurable_fst'' borel_measurable_times |
|
810 |
dest!: distributed_real_measurable) |
|
811 |
finally have mi_eq: |
|
812 |
"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))" . |
|
813 |
||
814 |
have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0" |
|
815 |
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto |
|
816 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
817 |
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) (auto intro: measurable_snd') |
|
818 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
819 |
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) (auto intro: measurable_Pair measurable_snd') |
|
820 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
|
821 |
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) (auto intro: measurable_Pair) |
|
822 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)" |
|
823 |
using Px by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
|
824 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)" |
|
825 |
using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
|
826 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))" |
|
827 |
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) |
|
828 |
moreover have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))" |
|
829 |
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair] |
|
830 |
using measurable_comp[OF measurable_Pair[OF measurable_fst measurable_comp[OF measurable_snd measurable_snd]] Pxz[THEN distributed_real_measurable], of T] |
|
831 |
using measurable_comp[OF measurable_snd measurable_Pair2[OF Pxz[THEN distributed_real_measurable]], of _ T] |
|
832 |
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def) |
|
833 |
moreover note Pxyz[THEN distributed_real_AE] |
|
834 |
ultimately have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. |
|
835 |
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) - |
|
836 |
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) = |
|
837 |
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) " |
|
838 |
proof eventually_elim |
|
839 |
case (goal1 x) |
|
840 |
show ?case |
|
40859 | 841 |
proof cases |
47694 | 842 |
assume "Pxyz x \<noteq> 0" |
843 |
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" |
|
844 |
by auto |
|
845 |
then show ?thesis |
|
846 |
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps) |
|
40859 | 847 |
qed simp |
848 |
qed |
|
47694 | 849 |
with I1 I2 show ?thesis |
40859 | 850 |
unfolding conditional_mutual_information_def |
47694 | 851 |
apply (subst mi_eq) |
852 |
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz]) |
|
853 |
apply (subst integral_diff(2)[symmetric]) |
|
854 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
|
855 |
done |
|
40859 | 856 |
qed |
857 |
||
858 |
lemma (in information_space) conditional_mutual_information_eq: |
|
47694 | 859 |
assumes Pz: "simple_distributed M Z Pz" |
860 |
assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz" |
|
861 |
assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz" |
|
862 |
assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
863 |
shows "\<I>(X ; Y | Z) = |
|
864 |
(\<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))))" |
|
865 |
proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _ |
|
866 |
simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz] |
|
867 |
simple_distributed_joint2[OF Pxyz]]) |
|
868 |
note simple_distributed_joint2_finite[OF Pxyz, simp] |
|
869 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
870 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
871 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
872 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
873 |
show "sigma_finite_measure (count_space (Z ` space M))" |
|
874 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
875 |
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) = |
|
876 |
count_space (X`space M \<times> Y`space M \<times> Z`space M)" |
|
877 |
(is "?P = ?C") |
|
878 |
by (simp add: pair_measure_count_space) |
|
40859 | 879 |
|
47694 | 880 |
let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)" |
881 |
have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^isub>M count_space (Z ` space M))" |
|
882 |
using simple_distributed_joint[OF Pxz] by (rule distributed_measurable) |
|
883 |
from measurable_comp[OF this measurable_fst] |
|
884 |
have "random_variable (count_space (X ` space M)) X" |
|
885 |
by (simp add: comp_def) |
|
886 |
then have "simple_function M X" |
|
887 |
unfolding simple_function_def by auto |
|
888 |
then have "simple_distributed M X ?Px" |
|
889 |
by (rule simple_distributedI) auto |
|
890 |
then show "distributed M (count_space (X ` space M)) X ?Px" |
|
891 |
by (rule simple_distributed) |
|
892 |
||
893 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)" |
|
894 |
let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)" |
|
895 |
let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)" |
|
896 |
show |
|
897 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))" |
|
898 |
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))" |
|
899 |
by (auto intro!: integrable_count_space simp: pair_measure_count_space) |
|
900 |
let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))" |
|
901 |
let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))" |
|
902 |
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)" |
|
903 |
by (auto intro!: ext) |
|
904 |
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)" |
|
905 |
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta') |
|
36624 | 906 |
qed |
907 |
||
47694 | 908 |
lemma (in information_space) conditional_mutual_information_nonneg: |
909 |
assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z" |
|
910 |
shows "0 \<le> \<I>(X ; Y | Z)" |
|
911 |
proof - |
|
912 |
def Pz \<equiv> "\<lambda>x. if x \<in> Z`space M then measure M (Z -` {x} \<inter> space M) else 0" |
|
913 |
def Pxz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x))`space M then measure M ((\<lambda>x. (X x, Z x)) -` {x} \<inter> space M) else 0" |
|
914 |
def Pyz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x))`space M then measure M ((\<lambda>x. (Y x, Z x)) -` {x} \<inter> space M) else 0" |
|
915 |
def Pxyz \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then measure M ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M) else 0" |
|
916 |
let ?M = "X`space M \<times> Y`space M \<times> Z`space M" |
|
36624 | 917 |
|
47694 | 918 |
note XZ = simple_function_Pair[OF X Z] |
919 |
note YZ = simple_function_Pair[OF Y Z] |
|
920 |
note XYZ = simple_function_Pair[OF X simple_function_Pair[OF Y Z]] |
|
921 |
have Pz: "simple_distributed M Z Pz" |
|
922 |
using Z by (rule simple_distributedI) (auto simp: Pz_def) |
|
923 |
have Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz" |
|
924 |
using XZ by (rule simple_distributedI) (auto simp: Pxz_def) |
|
925 |
have Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz" |
|
926 |
using YZ by (rule simple_distributedI) (auto simp: Pyz_def) |
|
927 |
have Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz" |
|
928 |
using XYZ by (rule simple_distributedI) (auto simp: Pxyz_def) |
|
40859 | 929 |
|
47694 | 930 |
{ fix z assume z: "z \<in> Z ` space M" then have "(\<Sum>x\<in>X ` space M. Pxz (x, z)) = Pz z" |
931 |
using distributed_marginal_eq_joint_simple[OF X Pz Pxz z] |
|
932 |
by (auto intro!: setsum_cong simp: Pxz_def) } |
|
933 |
note marginal1 = this |
|
40859 | 934 |
|
47694 | 935 |
{ fix z assume z: "z \<in> Z ` space M" then have "(\<Sum>y\<in>Y ` space M. Pyz (y, z)) = Pz z" |
936 |
using distributed_marginal_eq_joint_simple[OF Y Pz Pyz z] |
|
937 |
by (auto intro!: setsum_cong simp: Pyz_def) } |
|
938 |
note marginal2 = this |
|
939 |
||
940 |
have "- \<I>(X ; Y | Z) = - (\<Sum>(x, y, z) \<in> ?M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))" |
|
941 |
unfolding conditional_mutual_information_eq[OF Pz Pyz Pxz Pxyz] |
|
942 |
using X Y Z by (auto intro!: setsum_mono_zero_left simp: Pxyz_def simple_functionD) |
|
943 |
also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. Pxz (x, z) * (Pyz (y,z) / Pz z))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41833
diff
changeset
|
944 |
unfolding split_beta' |
36624 | 945 |
proof (rule log_setsum_divide) |
47694 | 946 |
show "?M \<noteq> {}" using not_empty by simp |
36624 | 947 |
show "1 < b" using b_gt_1 . |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
948 |
|
47694 | 949 |
show "finite ?M" using X Y Z by (auto simp: simple_functionD) |
40859 | 950 |
|
47694 | 951 |
then show "(\<Sum>x\<in>?M. Pxyz (fst x, fst (snd x), snd (snd x))) = 1" |
952 |
apply (subst Pxyz[THEN simple_distributed_setsum_space, symmetric]) |
|
953 |
apply simp |
|
954 |
apply (intro setsum_mono_zero_right) |
|
955 |
apply (auto simp: Pxyz_def) |
|
956 |
done |
|
957 |
let ?N = "(\<lambda>x. (X x, Y x, Z x)) ` space M" |
|
958 |
fix x assume x: "x \<in> ?M" |
|
959 |
let ?Q = "Pxyz (fst x, fst (snd x), snd (snd x))" |
|
960 |
let ?P = "Pxz (fst x, snd (snd x)) * (Pyz (fst (snd x), snd (snd x)) / Pz (snd (snd x)))" |
|
961 |
from x show "0 \<le> ?Q" "0 \<le> ?P" |
|
962 |
using Pxyz[THEN simple_distributed, THEN distributed_real_AE] |
|
963 |
using Pxz[THEN simple_distributed, THEN distributed_real_AE] |
|
964 |
using Pyz[THEN simple_distributed, THEN distributed_real_AE] |
|
965 |
using Pz[THEN simple_distributed, THEN distributed_real_AE] |
|
966 |
by (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg simp: AE_count_space Pxyz_def Pxz_def Pyz_def Pz_def) |
|
967 |
moreover assume "0 < ?Q" |
|
968 |
moreover have "AE x in count_space ?N. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
969 |
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz[THEN simple_distributed] Pz[THEN simple_distributed]]) (auto intro: measurable_snd') |
|
970 |
then have "\<And>x. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
971 |
by (auto simp: Pz_def Pxyz_def AE_count_space) |
|
972 |
moreover have "AE x in count_space ?N. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
973 |
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz[THEN simple_distributed] Pxz[THEN simple_distributed]]) (auto intro: measurable_Pair measurable_snd') |
|
974 |
then have "\<And>x. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0" |
|
975 |
by (auto simp: Pz_def Pxyz_def AE_count_space) |
|
976 |
moreover have "AE x in count_space ?N. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
|
977 |
by (intro subdensity_real[of snd, OF _ Pxyz[THEN simple_distributed] Pyz[THEN simple_distributed]]) (auto intro: measurable_Pair) |
|
978 |
then have "\<And>x. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0" |
|
979 |
by (auto simp: Pz_def Pxyz_def AE_count_space) |
|
980 |
ultimately show "0 < ?P" using x by (auto intro!: divide_pos_pos mult_pos_pos simp: less_le) |
|
40859 | 981 |
qed |
47694 | 982 |
also have "(\<Sum>(x, y, z) \<in> ?M. Pxz (x, z) * (Pyz (y,z) / Pz z)) = (\<Sum>z\<in>Z`space M. Pz z)" |
36624 | 983 |
apply (simp add: setsum_cartesian_product') |
984 |
apply (subst setsum_commute) |
|
985 |
apply (subst (2) setsum_commute) |
|
47694 | 986 |
apply (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric] marginal1 marginal2 |
36624 | 987 |
intro!: setsum_cong) |
47694 | 988 |
done |
989 |
also have "log b (\<Sum>z\<in>Z`space M. Pz z) = 0" |
|
990 |
using Pz[THEN simple_distributed_setsum_space] by simp |
|
40859 | 991 |
finally show ?thesis by simp |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
992 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
993 |
|
39097 | 994 |
subsection {* Conditional Entropy *} |
995 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
996 |
definition (in prob_space) |
47694 | 997 |
"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
|
998 |
|
40859 | 999 |
abbreviation (in information_space) |
1000 |
conditional_entropy_Pow ("\<H>'(_ | _')") where |
|
47694 | 1001 |
"\<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
|
1002 |
|
40859 | 1003 |
lemma (in information_space) conditional_entropy_generic_eq: |
47694 | 1004 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" |
1005 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1006 |
assumes Px: "distributed M S X Px" |
|
1007 |
assumes Py: "distributed M T Y Py" |
|
1008 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
1009 |
assumes I1: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
|
1010 |
assumes I2: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
|
1011 |
shows "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))" |
|
40859 | 1012 |
proof - |
47694 | 1013 |
interpret S: sigma_finite_measure S by fact |
1014 |
interpret T: sigma_finite_measure T by fact |
|
1015 |
interpret ST: pair_sigma_finite S T .. |
|
1016 |
have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" .. |
|
1017 |
||
1018 |
interpret Pxy: prob_space "density (S \<Otimes>\<^isub>M T) Pxy" |
|
1019 |
unfolding Pxy[THEN distributed_distr_eq_density, symmetric] |
|
1020 |
using Pxy[THEN distributed_measurable] by (rule prob_space_distr) |
|
1021 |
||
1022 |
from Py Pxy have distr_eq: "distr M T Y = |
|
1023 |
distr (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) T snd" |
|
1024 |
by (subst distr_distr[OF measurable_snd]) (auto dest: distributed_measurable simp: comp_def) |
|
1025 |
||
1026 |
have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)" |
|
1027 |
by (rule entropy_distr[OF T Py]) |
|
1028 |
also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" |
|
1029 |
using b_gt_1 Py[THEN distributed_real_measurable] |
|
1030 |
by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong) |
|
1031 |
finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" . |
|
1032 |
||
1033 |
have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
|
1034 |
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair) |
|
1035 |
moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1036 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
|
1037 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)" |
|
1038 |
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
|
1039 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)" |
|
1040 |
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
|
1041 |
moreover note Pxy[THEN distributed_real_AE] |
|
1042 |
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> |
|
1043 |
(Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Px (fst x) \<and> 0 < Py (snd x)))" |
|
1044 |
by eventually_elim auto |
|
1045 |
||
1046 |
from pos have "AE x in S \<Otimes>\<^isub>M T. |
|
1047 |
Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))" |
|
1048 |
by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1) |
|
1049 |
with I1 I2 show ?thesis |
|
40859 | 1050 |
unfolding conditional_entropy_def |
47694 | 1051 |
apply (subst e_eq) |
1052 |
apply (subst entropy_distr[OF ST Pxy]) |
|
1053 |
unfolding minus_diff_minus |
|
1054 |
apply (subst integral_diff(2)[symmetric]) |
|
1055 |
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff) |
|
1056 |
done |
|
39097 | 1057 |
qed |
1058 |
||
40859 | 1059 |
lemma (in information_space) conditional_entropy_eq: |
47694 | 1060 |
assumes Y: "simple_distributed M Y Py" and X: "simple_function M X" |
1061 |
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
1062 |
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))" |
|
1063 |
proof (subst conditional_entropy_generic_eq[OF _ _ |
|
1064 |
simple_distributed[OF simple_distributedI[OF X refl]] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
|
1065 |
have [simp]: "finite (X`space M)" using X by (simp add: simple_function_def) |
|
1066 |
note Y[THEN simple_distributed_finite, simp] |
|
1067 |
show "sigma_finite_measure (count_space (X ` space M))" |
|
1068 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1069 |
show "sigma_finite_measure (count_space (Y ` space M))" |
|
1070 |
by (simp add: sigma_finite_measure_count_space_finite) |
|
1071 |
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)" |
|
1072 |
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)" |
|
1073 |
(is "?P = ?C") |
|
1074 |
using X Y by (simp add: simple_distributed_finite pair_measure_count_space) |
|
1075 |
with X Y show |
|
1076 |
"integrable ?P (\<lambda>x. ?f x * log b (?f x))" |
|
1077 |
"integrable ?P (\<lambda>x. ?f x * log b (Py (snd x)))" |
|
1078 |
by (auto intro!: integrable_count_space simp: simple_distributed_finite) |
|
1079 |
have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) = |
|
1080 |
(\<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)" |
|
1081 |
by auto |
|
1082 |
from X Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) = |
|
1083 |
- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))" |
|
1084 |
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta') |
|
1085 |
qed |
|
39097 | 1086 |
|
47694 | 1087 |
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
|
1088 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
47694 | 1089 |
shows "\<I>(X ; X | Y) = \<H>(X | Y)" |
1090 |
proof - |
|
1091 |
def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0" |
|
1092 |
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" |
|
1093 |
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" |
|
1094 |
let ?M = "X`space M \<times> X`space M \<times> Y`space M" |
|
39097 | 1095 |
|
47694 | 1096 |
note XY = simple_function_Pair[OF X Y] |
1097 |
note XXY = simple_function_Pair[OF X XY] |
|
1098 |
have Py: "simple_distributed M Y Py" |
|
1099 |
using Y by (rule simple_distributedI) (auto simp: Py_def) |
|
1100 |
have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" |
|
1101 |
using XY by (rule simple_distributedI) (auto simp: Pxy_def) |
|
1102 |
have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy" |
|
1103 |
using XXY by (rule simple_distributedI) (auto simp: Pxxy_def) |
|
1104 |
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" |
|
1105 |
by auto |
|
1106 |
have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A" |
|
1107 |
by (auto simp: inj_on_def) |
|
1108 |
have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)" |
|
1109 |
by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob]) |
|
1110 |
have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1111 |
by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair) |
|
1112 |
then show ?thesis |
|
1113 |
apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy]) |
|
1114 |
apply (subst conditional_entropy_eq[OF Py X Pxy]) |
|
1115 |
apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj] |
|
1116 |
log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space) |
|
1117 |
using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE] |
|
1118 |
apply (auto simp add: not_le[symmetric] AE_count_space) |
|
1119 |
done |
|
1120 |
qed |
|
1121 |
||
1122 |
lemma (in information_space) conditional_entropy_nonneg: |
|
1123 |
assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)" |
|
1124 |
using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y] |
|
1125 |
by simp |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1126 |
|
39097 | 1127 |
subsection {* Equalities *} |
1128 |
||
47694 | 1129 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr: |
1130 |
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real" |
|
1131 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1132 |
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py" |
|
1133 |
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy" |
|
1134 |
assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))" |
|
1135 |
assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" |
|
1136 |
assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" |
|
1137 |
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 | 1138 |
proof - |
47694 | 1139 |
have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^isub>M T))" |
1140 |
using b_gt_1 Px[THEN distributed_real_measurable] |
|
1141 |
apply (subst entropy_distr[OF S Px]) |
|
1142 |
apply (subst distributed_transform_integral[OF Pxy Px, where T=fst]) |
|
1143 |
apply (auto intro!: integral_cong) |
|
1144 |
done |
|
1145 |
||
1146 |
have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))" |
|
1147 |
using b_gt_1 Py[THEN distributed_real_measurable] |
|
1148 |
apply (subst entropy_distr[OF T Py]) |
|
1149 |
apply (subst distributed_transform_integral[OF Pxy Py, where T=snd]) |
|
1150 |
apply (auto intro!: integral_cong) |
|
1151 |
done |
|
1152 |
||
1153 |
interpret S: sigma_finite_measure S by fact |
|
1154 |
interpret T: sigma_finite_measure T by fact |
|
1155 |
interpret ST: pair_sigma_finite S T .. |
|
1156 |
have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" .. |
|
1157 |
||
1158 |
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))" |
|
1159 |
by (subst entropy_distr[OF ST Pxy]) (auto intro!: integral_cong) |
|
1160 |
||
1161 |
have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0" |
|
1162 |
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair) |
|
1163 |
moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0" |
|
1164 |
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair) |
|
1165 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)" |
|
1166 |
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable) |
|
1167 |
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)" |
|
1168 |
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable) |
|
1169 |
moreover note Pxy[THEN distributed_real_AE] |
|
1170 |
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)) = |
|
1171 |
Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))" |
|
1172 |
(is "AE x in _. ?f x = ?g x") |
|
1173 |
proof eventually_elim |
|
1174 |
case (goal1 x) |
|
1175 |
show ?case |
|
1176 |
proof cases |
|
1177 |
assume "Pxy x \<noteq> 0" |
|
1178 |
with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x" |
|
1179 |
by auto |
|
1180 |
then show ?thesis |
|
1181 |
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps) |
|
1182 |
qed simp |
|
1183 |
qed |
|
1184 |
||
1185 |
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" |
|
1186 |
unfolding X Y XY |
|
1187 |
apply (subst integral_diff) |
|
1188 |
apply (intro integral_diff Ixy Ix Iy)+ |
|
1189 |
apply (subst integral_diff) |
|
1190 |
apply (intro integral_diff Ixy Ix Iy)+ |
|
1191 |
apply (simp add: field_simps) |
|
1192 |
done |
|
1193 |
also have "\<dots> = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?g" |
|
1194 |
using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE) |
|
1195 |
also have "\<dots> = mutual_information b S T X Y" |
|
1196 |
by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric]) |
|
1197 |
finally show ?thesis .. |
|
1198 |
qed |
|
1199 |
||
1200 |
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy: |
|
1201 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1202 |
shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)" |
|
1203 |
proof - |
|
1204 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1205 |
using sf_X by (rule simple_distributedI) auto |
|
1206 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1207 |
using sf_Y by (rule simple_distributedI) auto |
|
1208 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1209 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1210 |
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))" |
|
1211 |
by (rule simple_distributedI) auto |
|
1212 |
from simple_distributed_joint_finite[OF this, simp] |
|
1213 |
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
|
1214 |
by (simp add: pair_measure_count_space) |
|
1215 |
||
1216 |
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))" |
|
1217 |
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] |
|
1218 |
by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space) |
|
1219 |
then show ?thesis |
|
1220 |
unfolding conditional_entropy_def by simp |
|
1221 |
qed |
|
1222 |
||
1223 |
lemma (in information_space) mutual_information_nonneg_simple: |
|
1224 |
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y" |
|
1225 |
shows "0 \<le> \<I>(X ; Y)" |
|
1226 |
proof - |
|
1227 |
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))" |
|
1228 |
using sf_X by (rule simple_distributedI) auto |
|
1229 |
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))" |
|
1230 |
using sf_Y by (rule simple_distributedI) auto |
|
1231 |
||
1232 |
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))" |
|
1233 |
using sf_X sf_Y by (rule simple_function_Pair) |
|
1234 |
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))" |
|
1235 |
by (rule simple_distributedI) auto |
|
1236 |
||
1237 |
from simple_distributed_joint_finite[OF this, simp] |
|
1238 |
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)" |
|
1239 |
by (simp add: pair_measure_count_space) |
|
1240 |
||
40859 | 1241 |
show ?thesis |
47694 | 1242 |
by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]]) |
1243 |
(simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite) |
|
40859 | 1244 |
qed |
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1245 |
|
40859 | 1246 |
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
|
1247 |
assumes X: "simple_function M X" and Z: "simple_function M Z" |
40859 | 1248 |
shows "\<H>(X | Z) \<le> \<H>(X)" |
36624 | 1249 |
proof - |
47694 | 1250 |
have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple) |
1251 |
also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] . |
|
1252 |
finally show ?thesis by auto |
|
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1253 |
qed |
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1254 |
|
40859 | 1255 |
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
|
1256 |
assumes X: "simple_function M X" and Y: "simple_function M Y" |
40859 | 1257 |
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)" |
1258 |
proof - |
|
47694 | 1259 |
note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl] |
1260 |
note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl] |
|
1261 |
note simple_distributed_joint_finite[OF this, simp] |
|
1262 |
let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" |
|
1263 |
let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)" |
|
1264 |
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" |
|
1265 |
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))" |
|
1266 |
using XY by (rule entropy_simple_distributed) |
|
1267 |
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))" |
|
1268 |
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)"]) |
|
1269 |
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))" |
|
1270 |
by (auto intro!: setsum_cong) |
|
1271 |
also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" |
|
1272 |
by (subst entropy_distr[OF _ simple_distributed_joint[OF YX]]) |
|
1273 |
(auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite |
|
1274 |
cong del: setsum_cong intro!: setsum_mono_zero_left) |
|
1275 |
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))" . |
|
1276 |
then show ?thesis |
|
1277 |
unfolding conditional_entropy_def by simp |
|
36624 | 1278 |
qed |
1279 |
||
40859 | 1280 |
lemma (in information_space) entropy_partition: |
47694 | 1281 |
assumes X: "simple_function M X" |
1282 |
shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)" |
|
36624 | 1283 |
proof - |
47694 | 1284 |
note fX = simple_function_compose[OF X, of f] |
1285 |
have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto |
|
1286 |
have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A" |
|
1287 |
by (auto simp: inj_on_def) |
|
1288 |
show ?thesis |
|
1289 |
apply (subst entropy_chain_rule[symmetric, OF fX X]) |
|
1290 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]]) |
|
1291 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1292 |
unfolding eq |
|
1293 |
apply (subst setsum_reindex[OF inj]) |
|
1294 |
apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"]) |
|
1295 |
done |
|
36624 | 1296 |
qed |
1297 |
||
40859 | 1298 |
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
|
1299 |
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)" |
40859 | 1300 |
proof - |
47694 | 1301 |
note fX = simple_function_compose[OF X, of f] |
1302 |
from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition) |
|
40859 | 1303 |
then show "\<H>(f \<circ> X) \<le> \<H>(X)" |
47694 | 1304 |
by (auto intro: conditional_entropy_nonneg[OF X fX]) |
40859 | 1305 |
qed |
36624 | 1306 |
|
40859 | 1307 |
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
|
1308 |
assumes X: "simple_function M X" and inj: "inj_on f (X`space M)" |
36624 | 1309 |
shows "\<H>(f \<circ> X) = \<H>(X)" |
1310 |
proof (rule antisym) |
|
40859 | 1311 |
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] . |
36624 | 1312 |
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
|
1313 |
have sf: "simple_function M (f \<circ> X)" |
40859 | 1314 |
using X by auto |
36624 | 1315 |
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))" |
47694 | 1316 |
unfolding o_assoc |
1317 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]]) |
|
1318 |
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"]) |
|
1319 |
apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def) |
|
1320 |
done |
|
36624 | 1321 |
also have "... \<le> \<H>(f \<circ> X)" |
40859 | 1322 |
using entropy_data_processing[OF sf] . |
36624 | 1323 |
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" . |
1324 |
qed |
|
1325 |
||
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset
|
1326 |
end |