src/HOL/Probability/Information.thy
author hoelzl
Tue, 22 Mar 2011 20:06:10 +0100
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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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  Probability_Space
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  "~~/src/HOL/Library/Convex"
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begin
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lemma (in prob_space) not_zero_less_distribution[simp]:
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  "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
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  using distribution_positive[of X A] by arith
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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 fastsimp
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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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    87
  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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   105
  proof (rule log_mono)
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   106
    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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   108
    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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   146
    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)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   152
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   153
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   154
lemma log_mult_eq:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   155
  "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   156
  using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   157
  by (auto simp: zero_less_mult_iff mult_le_0_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   158
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   159
lemma log_inverse_eq:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   160
  "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   161
  using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   162
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   163
lemma log_divide_eq:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   164
  "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   165
  unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   166
  by (auto simp: zero_less_mult_iff mult_le_0_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   167
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   168
lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   169
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   170
end
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   171
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   172
subsection "Kullback$-$Leibler divergence"
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   173
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   174
text {* The Kullback$-$Leibler divergence is also known as relative entropy or
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   175
Kullback$-$Leibler distance. *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   176
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   177
definition
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
   178
  "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv M \<nu> x)) \<partial>M\<lparr>measure := \<nu>\<rparr>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   179
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   180
lemma (in sigma_finite_measure) KL_divergence_vimage:
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   181
  assumes T: "T \<in> measure_preserving M M'"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   182
    and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   183
    and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   184
    and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   185
  and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   186
  and "1 < b"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   187
  shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   188
proof -
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   189
  interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   190
  have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   191
    by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   192
  have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   193
  then have saM': "sigma_algebra M'" by simp
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   194
  then interpret M': measure_space M' by (rule measure_space_vimage) fact
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   195
  have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   196
  proof safe
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   197
    fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   198
    then have N': "T' -` N \<inter> space M' \<in> sets M'"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   199
      using T' by (auto simp: measurable_def measure_preserving_def)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   200
    have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   201
      using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   202
    then have "measure M' (T' -` N \<inter> space M') = 0"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   203
      using measure_preservingD[OF T N'] N_0 by auto
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   204
    with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   205
      unfolding M'.absolutely_continuous_def measurable_def by auto
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   206
  qed
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   207
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   208
  have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   209
  have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   210
    by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   211
  show ?thesis
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   212
    unfolding KL_divergence_def
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   213
  proof (subst \<nu>'.integral_vimage[OF sa T'])
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   214
    show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   215
      by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   216
    have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   217
      (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   218
      using inv' by (auto intro!: \<nu>'.integral_cong)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   219
    also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   220
      using M ac AE
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   221
      by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   222
         (auto elim!: AE_mp)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   223
    finally show "?l = ?r" .
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   224
  qed
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   225
qed
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   226
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   227
lemma (in sigma_finite_measure) KL_divergence_cong:
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
   228
  assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
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
   229
  assumes [simp]: "sets N = sets M" "space N = space M"
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
   230
    "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
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
   231
    "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
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
   232
  shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   233
proof -
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
   234
  interpret \<nu>: measure_space ?\<nu> by fact
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
   235
  have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
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
   236
    by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def)
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
   237
  also have "\<dots> = KL_divergence b N \<nu>'"
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
   238
    by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
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
   239
  finally show ?thesis .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   240
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   241
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   242
lemma (in finite_measure_space) KL_divergence_eq_finite:
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
   243
  assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   244
  assumes ac: "absolutely_continuous \<nu>"
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
   245
  shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   246
proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
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
   247
  interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
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
   248
  have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
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
   249
  show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   250
    using RN_deriv_finite_measure[OF ms ac]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   251
    by (auto intro!: setsum_cong simp: field_simps)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   252
qed
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   253
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   254
lemma (in finite_prob_space) KL_divergence_positive_finite:
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
   255
  assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   256
  assumes ac: "absolutely_continuous \<nu>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   257
  and "1 < b"
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
   258
  shows "0 \<le> KL_divergence b M \<nu>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   259
proof -
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
   260
  interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
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
   261
  have ms: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   262
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
   263
  have "- (KL_divergence b M \<nu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   264
  proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   265
    show "finite (space M)" using finite_space by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   266
    show "1 < b" by fact
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   267
    show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   268
      using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   269
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   270
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   271
    then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   272
    { assume "0 < real (\<nu> {x})"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   273
      then have "\<nu> {x} \<noteq> 0" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   274
      then have "\<mu> {x} \<noteq> 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   275
        using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   276
      thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   277
    show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   278
      using real_measure[OF x] v.real_measure[of "{x}"] x by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   279
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   280
  thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   281
qed
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   282
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   283
subsection {* Mutual Information *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   284
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   285
definition (in prob_space)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   286
  "mutual_information b S T X Y =
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   287
    KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   288
      (extreal\<circ>joint_distribution X Y)"
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   289
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   290
definition (in prob_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   291
  "entropy b s X = mutual_information b s s X X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   292
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   293
abbreviation (in information_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   294
  mutual_information_Pow ("\<I>'(_ ; _')") where
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   295
  "\<I>(X ; Y) \<equiv> mutual_information b
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   296
    \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   297
    \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> 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
   298
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   299
lemma (in prob_space) finite_variables_absolutely_continuous:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   300
  assumes X: "finite_random_variable S X" and Y: "finite_random_variable T 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
   301
  shows "measure_space.absolutely_continuous
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   302
    (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   303
    (extreal\<circ>joint_distribution X Y)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   304
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   305
  interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
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
   306
    using X by (rule distribution_finite_prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   307
  interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
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
   308
    using Y by (rule distribution_finite_prob_space)
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
   309
  interpret XY: pair_finite_prob_space
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   310
    "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   311
  interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
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
   312
    using assms by (auto intro!: joint_distribution_finite_prob_space)
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
   313
  note rv = assms[THEN finite_random_variableD]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   314
  show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   315
  proof (rule XY.absolutely_continuousI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   316
    show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
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
   317
    fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   318
    then obtain a b where "x = (a, b)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   319
      and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
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
   320
      by (cases x) (auto simp: space_pair_measure)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   321
    with finite_distribution_order(5,6)[OF X Y]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   322
    show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   323
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   324
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   325
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   326
lemma (in information_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   327
  assumes MX: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   328
  assumes MY: "finite_random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   329
  shows mutual_information_generic_eq:
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   330
    "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   331
      joint_distribution X Y {(x,y)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   332
      log b (joint_distribution X Y {(x,y)} /
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   333
      (distribution X {x} * distribution Y {y})))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   334
    (is ?sum)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   335
  and mutual_information_positive_generic:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   336
     "0 \<le> mutual_information b MX MY X Y" (is ?positive)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   337
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   338
  interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
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
   339
    using MX by (rule distribution_finite_prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   340
  interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
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
   341
    using MY by (rule distribution_finite_prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   342
  interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   343
  interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
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
   344
    using assms by (auto intro!: joint_distribution_finite_prob_space)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   345
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   346
  have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   347
  have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   348
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   349
  show ?sum
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   350
    unfolding Let_def mutual_information_def
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   351
    by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   352
       (auto simp add: space_pair_measure setsum_cartesian_product')
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   353
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   354
  show ?positive
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   355
    using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   356
    unfolding mutual_information_def .
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   357
qed
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   358
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   359
lemma (in information_space) mutual_information_commute_generic:
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   360
  assumes X: "random_variable S X" and Y: "random_variable T Y"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   361
  assumes ac: "measure_space.absolutely_continuous
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   362
    (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   363
  shows "mutual_information b S T X Y = mutual_information b T S Y X"
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   364
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   365
  let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   366
  interpret S: prob_space ?S using X by (rule distribution_prob_space)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   367
  interpret T: prob_space ?T using Y by (rule distribution_prob_space)
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   368
  interpret P: pair_prob_space ?S ?T ..
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   369
  interpret Q: pair_prob_space ?T ?S ..
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   370
  show ?thesis
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   371
    unfolding mutual_information_def
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   372
  proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   373
    show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   374
      (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   375
      using X Y unfolding measurable_def
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   376
      unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   377
      by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   378
    have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   379
      using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   380
    then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   381
      unfolding prob_space_def by simp
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   382
  qed auto
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   383
qed
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   384
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   385
lemma (in information_space) mutual_information_commute:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   386
  assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   387
  shows "mutual_information b S T X Y = mutual_information b T S Y X"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   388
  unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   389
  unfolding joint_distribution_commute_singleton[of X Y]
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   390
  by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   391
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   392
lemma (in information_space) mutual_information_commute_simple:
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
   393
  assumes X: "simple_function M X" and Y: "simple_function M Y"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   394
  shows "\<I>(X;Y) = \<I>(Y;X)"
41833
563bea92b2c0 add lemma KL_divergence_vimage, mutual_information_generic
hoelzl
parents: 41689
diff changeset
   395
  by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41413
diff changeset
   396
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   397
lemma (in information_space) mutual_information_eq:
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
   398
  assumes "simple_function M X" "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   399
  shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   400
    distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   401
                                                   (distribution X {x} * distribution Y {y})))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   402
  using assms by (simp add: mutual_information_generic_eq)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   403
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   404
lemma (in information_space) mutual_information_generic_cong:
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   405
  assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   406
  assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   407
  shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   408
  unfolding mutual_information_def using X Y
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   409
  by (simp cong: distribution_cong)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   410
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   411
lemma (in information_space) mutual_information_cong:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   412
  assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   413
  assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   414
  shows "\<I>(X; Y) = \<I>(X'; Y')"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   415
  unfolding mutual_information_def using X Y
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   416
  by (simp cong: distribution_cong image_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   417
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   418
lemma (in information_space) mutual_information_positive:
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
   419
  assumes "simple_function M X" "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   420
  shows "0 \<le> \<I>(X;Y)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   421
  using assms by (simp add: mutual_information_positive_generic)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   422
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   423
subsection {* Entropy *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   424
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   425
abbreviation (in information_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   426
  entropy_Pow ("\<H>'(_')") where
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   427
  "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   428
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   429
lemma (in information_space) entropy_generic_eq:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   430
  fixes X :: "'a \<Rightarrow> 'c"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   431
  assumes MX: "finite_random_variable MX X"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   432
  shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   433
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   434
  interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
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
   435
    using MX by (rule distribution_finite_prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   436
  let "?X x" = "distribution X {x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   437
  let "?XX x y" = "joint_distribution X X {(x, y)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   438
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   439
  { fix x y :: 'c
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   440
    { assume "x \<noteq> y"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   441
      then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   442
      then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   443
    then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   444
        (if x = y then - ?X y * log b (?X y) else 0)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   445
      by (auto simp: log_simps zero_less_mult_iff) }
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   446
  note remove_XX = this
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   447
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   448
  show ?thesis
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   449
    unfolding entropy_def mutual_information_generic_eq[OF MX MX]
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   450
    unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
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
   451
    using MX.finite_space by (auto simp: setsum_cases)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   452
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   453
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   454
lemma (in information_space) entropy_eq:
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
   455
  assumes "simple_function M X"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   456
  shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   457
  using assms by (simp add: entropy_generic_eq)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   458
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   459
lemma (in information_space) entropy_positive:
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
   460
  "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   461
  unfolding entropy_def by (simp add: mutual_information_positive)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   462
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   463
lemma (in information_space) entropy_certainty_eq_0:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   464
  assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   465
  shows "\<H>(X) = 0"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   466
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   467
  let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
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
   468
  note simple_function_imp_finite_random_variable[OF `simple_function M X`]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   469
  from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
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
   470
  interpret X: finite_prob_space ?X by simp
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   471
  have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   472
    using X.measure_compl[of "{x}"] assms by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   473
  also have "\<dots> = 0" using X.prob_space assms by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   474
  finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   475
  { fix y assume *: "y \<in> X ` space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   476
    { assume asm: "y \<noteq> x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   477
      with * have "{y} \<subseteq> X ` space M - {x}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   478
      from X.measure_mono[OF this] X0 asm *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   479
      have "distribution X {y} = 0"  by (auto intro: antisym) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   480
    then have "distribution X {y} = (if x = y then 1 else 0)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   481
      using assms by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   482
  note fi = this
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   483
  have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
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
   484
  show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   485
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   486
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   487
lemma (in information_space) entropy_le_card_not_0:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   488
  assumes X: "simple_function M X"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   489
  shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   490
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   491
  let "?p x" = "distribution X {x}"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   492
  have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   493
    unfolding entropy_eq[OF X] setsum_negf[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   494
    by (auto intro!: setsum_cong simp: log_simps)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   495
  also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   496
    using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   497
    by (intro log_setsum') (auto simp: simple_function_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   498
  also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   499
    by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   500
  finally show ?thesis
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
   501
    using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   502
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   503
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   504
lemma (in prob_space) measure'_translate:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   505
  assumes X: "random_variable S X" and A: "A \<in> sets S"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   506
  shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   507
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   508
  interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   509
    using distribution_prob_space[OF X] .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   510
  from A show "S.\<mu>' A = distribution X A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   511
    unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   512
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   513
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   514
lemma (in information_space) entropy_uniform_max:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   515
  assumes X: "simple_function M X"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   516
  assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   517
  shows "\<H>(X) = log b (real (card (X ` space M)))"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   518
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   519
  let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   520
  note frv = simple_function_imp_finite_random_variable[OF X]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   521
  from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
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
   522
  interpret X: finite_prob_space ?X by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   523
  note rv = finite_random_variableD[OF frv]
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   524
  have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
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
   525
    using `simple_function M X` not_empty by (auto simp: simple_function_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   526
  { fix x assume "x \<in> space ?X"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   527
    moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   528
    proof (rule X.uniform_prob)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   529
      fix x y assume "x \<in> space ?X" "y \<in> space ?X"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   530
      with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   531
        by (subst (1 2) measure'_translate[OF rv]) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   532
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   533
    ultimately have "distribution X {x} = 1 / card (space ?X)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   534
      by (subst (asm) measure'_translate[OF rv]) auto }
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   535
  thus ?thesis
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   536
    using not_empty X.finite_space b_gt_1 card_gt0
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
   537
    by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   538
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   539
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   540
lemma (in information_space) entropy_le_card:
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
   541
  assumes "simple_function M X"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   542
  shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   543
proof cases
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   544
  assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   545
  then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   546
  moreover
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   547
  have "0 < card (X`space M)"
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
   548
    using `simple_function M X` not_empty
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   549
    by (auto simp: card_gt_0_iff simple_function_def)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   550
  then have "log b 1 \<le> log b (real (card (X`space M)))"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   551
    using b_gt_1 by (intro log_le) auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   552
  ultimately show ?thesis using assms by (simp add: entropy_eq)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   553
next
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   554
  assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   555
  have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   556
    (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   557
  note entropy_le_card_not_0[OF assms]
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   558
  also have "log b (real ?A) \<le> log b (real ?B)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   559
    using b_gt_1 False not_empty `?A \<le> ?B` assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   560
    by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   561
  finally show ?thesis .
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   562
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   563
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   564
lemma (in information_space) entropy_commute:
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
   565
  assumes "simple_function M X" "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   566
  shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   567
proof -
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
   568
  have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   569
    using assms by (auto intro: simple_function_Pair)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   570
  have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   571
    by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   572
  have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   573
    by (auto intro!: inj_onI)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   574
  show ?thesis
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   575
    unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   576
    by (simp add: joint_distribution_commute[of Y X] split_beta)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   577
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   578
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   579
lemma (in information_space) entropy_eq_cartesian_product:
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
   580
  assumes "simple_function M X" "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   581
  shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   582
    joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   583
proof -
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
   584
  have sf: "simple_function M (\<lambda>x. (X x, Y x))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   585
    using assms by (auto intro: simple_function_Pair)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   586
  { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   587
    then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   588
    then have "joint_distribution X Y {x} = 0"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   589
      unfolding distribution_def by auto }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   590
  then show ?thesis using sf assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   591
    unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   592
    by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   593
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   594
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   595
subsection {* Conditional Mutual Information *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   596
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   597
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
   598
  "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
   599
    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
   600
    mutual_information b MX MZ X Z"
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   601
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   602
abbreviation (in information_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   603
  conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   604
  "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   605
    \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   606
    \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   607
    \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   608
    X Y Z"
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   609
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   610
lemma (in information_space) conditional_mutual_information_generic_eq:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   611
  assumes MX: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   612
    and MY: "finite_random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   613
    and MZ: "finite_random_variable MZ Z"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   614
  shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   615
             distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   616
             log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   617
    (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   618
  (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   619
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   620
  let ?X = "\<lambda>x. distribution X {x}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   621
  note finite_var = MX MY MZ
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   622
  note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   623
  note XYZ = finite_random_variable_pairI[OF MX YZ]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   624
  note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   625
  note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   626
  note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   627
  note order1 =
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   628
    finite_distribution_order(5,6)[OF finite_var(1) YZ]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   629
    finite_distribution_order(5,6)[OF finite_var(1,3)]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   630
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   631
  note random_var = finite_var[THEN finite_random_variableD]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   632
  note finite = finite_var(1) YZ finite_var(3) XZ YZX
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   633
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   634
  have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   635
          \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   636
    unfolding joint_distribution_commute_singleton[of X]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   637
    unfolding joint_distribution_assoc_singleton[symmetric]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   638
    using finite_distribution_order(6)[OF finite_var(2) ZX]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   639
    by auto
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   640
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   641
  have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   642
    (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   643
    (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   644
  proof (safe intro!: setsum_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   645
    fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   646
    show "?L x y z = ?R x y z"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   647
    proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   648
      assume "?XYZ x y z \<noteq> 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   649
      with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   650
        using order1 order2 by (auto simp: less_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   651
      with b_gt_1 show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   652
        by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   653
    qed simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   654
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   655
  also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   656
                  (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   657
    by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   658
  also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   659
             (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   660
    unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   661
              setsum_left_distrib[symmetric]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   662
    unfolding joint_distribution_commute_singleton[of X]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   663
    unfolding joint_distribution_assoc_singleton[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   664
    using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   665
    by (intro setsum_cong refl) (simp add: space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   666
  also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   667
             (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   668
             conditional_mutual_information b MX MY MZ X Y Z"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   669
    unfolding conditional_mutual_information_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   670
    unfolding mutual_information_generic_eq[OF finite_var(1,3)]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   671
    unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
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
   672
    by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   673
  finally show ?thesis by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   674
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   675
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   676
lemma (in information_space) conditional_mutual_information_eq:
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
   677
  assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   678
  shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   679
             distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   680
             log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   681
    (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   682
  by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   683
     simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   684
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   685
lemma (in information_space) conditional_mutual_information_eq_mutual_information:
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
   686
  assumes X: "simple_function M X" and Y: "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   687
  shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   688
proof -
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   689
  have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
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
   690
  have C: "simple_function M (\<lambda>x. ())" by auto
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   691
  show ?thesis
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   692
    unfolding conditional_mutual_information_eq[OF X Y C]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   693
    unfolding mutual_information_eq[OF X Y]
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   694
    by (simp add: setsum_cartesian_product' distribution_remove_const)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   695
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   696
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   697
lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   698
  unfolding distribution_def using prob_space by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   699
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   700
lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   701
  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   702
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   703
lemma (in prob_space) setsum_distribution:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   704
  assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   705
  using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   706
  using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   707
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   708
lemma (in prob_space) setsum_real_distribution:
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
   709
  fixes MX :: "('c, 'd) measure_space_scheme"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   710
  assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   711
  using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   712
  using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   713
  by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   714
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   715
lemma (in information_space) conditional_mutual_information_generic_positive:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   716
  assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   717
  shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   718
proof (cases "space MX \<times> space MY \<times> space MZ = {}")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   719
  case True show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   720
    unfolding conditional_mutual_information_generic_eq[OF assms] True
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   721
    by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   722
next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   723
  case False
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   724
  let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   725
  let ?dXZ = "joint_distribution X Z"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   726
  let ?dYZ = "joint_distribution Y Z"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   727
  let ?dX = "distribution X"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   728
  let ?dZ = "distribution Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   729
  let ?M = "space MX \<times> space MY \<times> space MZ"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   730
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   731
  note YZ = finite_random_variable_pairI[OF Y Z]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   732
  note XZ = finite_random_variable_pairI[OF X Z]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   733
  note ZX = finite_random_variable_pairI[OF Z X]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   734
  note YZ = finite_random_variable_pairI[OF Y Z]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   735
  note XYZ = finite_random_variable_pairI[OF X YZ]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   736
  note finite = Z YZ XZ XYZ
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   737
  have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   738
          \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   739
    unfolding joint_distribution_commute_singleton[of X]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   740
    unfolding joint_distribution_assoc_singleton[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   741
    using finite_distribution_order(6)[OF Y ZX]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   742
    by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   743
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   744
  note order = order
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   745
    finite_distribution_order(5,6)[OF X YZ]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   746
    finite_distribution_order(5,6)[OF Y Z]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   747
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   748
  have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   749
    log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   750
    unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   751
  also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   752
    unfolding split_beta'
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   753
  proof (rule log_setsum_divide)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   754
    show "?M \<noteq> {}" using False by simp
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   755
    show "1 < b" using b_gt_1 .
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   756
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   757
    show "finite ?M" using assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   758
      unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   759
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   760
    show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   761
      unfolding setsum_cartesian_product'
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   762
      unfolding setsum_commute[of _ "space MY"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   763
      unfolding setsum_commute[of _ "space MZ"]
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
   764
      by (simp_all add: space_pair_measure
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   765
                        setsum_joint_distribution_singleton[OF X YZ]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   766
                        setsum_joint_distribution_singleton[OF Y Z]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   767
                        setsum_distribution[OF Z])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   768
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   769
    fix x assume "x \<in> ?M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   770
    let ?x = "(fst x, fst (snd x), snd (snd x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   771
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   772
    show "0 \<le> ?dXYZ {?x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   773
      "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   774
     by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   775
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   776
    assume *: "0 < ?dXYZ {?x}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   777
    with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   778
      by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   779
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   780
  also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   781
    apply (simp add: setsum_cartesian_product')
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   782
    apply (subst setsum_commute)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   783
    apply (subst (2) setsum_commute)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   784
    by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   785
                   setsum_joint_distribution_singleton[OF X Z]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   786
                   setsum_joint_distribution_singleton[OF Y Z]
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   787
          intro!: setsum_cong)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   788
  also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   789
    unfolding setsum_real_distribution[OF Z] by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   790
  finally show ?thesis by simp
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   791
qed
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   792
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   793
lemma (in information_space) conditional_mutual_information_positive:
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
   794
  assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   795
  shows "0 \<le> \<I>(X;Y|Z)"
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
   796
  by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   797
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   798
subsection {* Conditional Entropy *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   799
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   800
definition (in prob_space)
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   801
  "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   802
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   803
abbreviation (in information_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   804
  conditional_entropy_Pow ("\<H>'(_ | _')") where
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   805
  "\<H>(X | Y) \<equiv> conditional_entropy b
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   806
    \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   807
    \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   808
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   809
lemma (in information_space) conditional_entropy_positive:
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
   810
  "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   811
  unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   812
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   813
lemma (in information_space) conditional_entropy_generic_eq:
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
   814
  fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   815
  assumes MX: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   816
  assumes MZ: "finite_random_variable MZ Z"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   817
  shows "conditional_entropy b MX MZ X Z =
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   818
     - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   819
         joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   820
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   821
  interpret MX: finite_sigma_algebra MX using MX by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   822
  interpret MZ: finite_sigma_algebra MZ using MZ by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   823
  let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   824
  let "?XZ x z" = "joint_distribution X Z {(x, z)}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   825
  let "?Z z" = "distribution Z {z}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   826
  let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   827
  { fix x z have "?XXZ x x z = ?XZ x z"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   828
      unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   829
  note this[simp]
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
   830
  { fix x x' :: 'c and z assume "x' \<noteq> x"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   831
    then have "?XXZ x x' z = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   832
      by (auto simp: distribution_def empty_measure'[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   833
               simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   834
  note this[simp]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   835
  { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   836
    then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   837
      = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   838
      by (auto intro!: setsum_cong)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   839
    also have "\<dots> = ?XZ x z * ?f x x z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   840
      using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   841
    also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   842
    also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   843
      using finite_distribution_order(6)[OF MX MZ]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   844
      by (auto simp: log_simps field_simps zero_less_mult_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   845
    finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   846
  note * = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   847
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   848
    unfolding conditional_entropy_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   849
    unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   850
    by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   851
                   setsum_commute[of _ "space MZ"] *
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   852
             intro!: setsum_cong)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   853
qed
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   854
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   855
lemma (in information_space) conditional_entropy_eq:
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
   856
  assumes "simple_function M X" "simple_function M Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   857
  shows "\<H>(X | Z) =
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   858
     - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   859
         joint_distribution X Z {(x, z)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   860
         log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   861
  by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   862
     simp
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   863
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   864
lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
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
   865
  assumes X: "simple_function M X" and Y: "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   866
  shows "\<H>(X | Y) =
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   867
    -(\<Sum>y\<in>Y`space M. distribution Y {y} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   868
      (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   869
              log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   870
  unfolding conditional_entropy_eq[OF assms]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   871
  using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   872
  by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   873
           intro!: setsum_cong)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   874
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   875
lemma (in information_space) conditional_entropy_eq_cartesian_product:
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
   876
  assumes "simple_function M X" "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   877
  shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   878
    joint_distribution X Y {(x,y)} *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   879
    log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   880
  unfolding conditional_entropy_eq[OF assms]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   881
  by (auto intro!: setsum_cong simp: setsum_cartesian_product')
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   882
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   883
subsection {* Equalities *}
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   884
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   885
lemma (in information_space) mutual_information_eq_entropy_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
   886
  assumes X: "simple_function M X" and Z: "simple_function M Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   887
  shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   888
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   889
  let "?XZ x z" = "joint_distribution X Z {(x, z)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   890
  let "?Z z" = "distribution Z {z}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   891
  let "?X x" = "distribution X {x}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   892
  note fX = X[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   893
  note fZ = Z[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   894
  note finite_distribution_order[OF fX fZ, simp]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   895
  { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   896
    have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   897
          ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   898
      by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   899
  note * = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   900
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   901
    unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   902
    using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   903
    by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   904
                     setsum_distribution)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   905
qed
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   906
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   907
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
   908
  assumes X: "simple_function M X" and Z: "simple_function M Z"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   909
  shows "\<H>(X | Z) \<le> \<H>(X)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   910
proof -
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   911
  have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   912
  with mutual_information_positive[OF X Z] entropy_positive[OF X]
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   913
  show ?thesis by auto
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   914
qed
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   915
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   916
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
   917
  assumes X: "simple_function M X" and Y: "simple_function M Y"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   918
  shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   919
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   920
  let "?XY x y" = "joint_distribution X Y {(x, y)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   921
  let "?Y y" = "distribution Y {y}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   922
  let "?X x" = "distribution X {x}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   923
  note fX = X[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   924
  note fY = Y[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   925
  note finite_distribution_order[OF fX fY, simp]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   926
  { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   927
    have "?XY x y * log b (?XY x y / ?X x) =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   928
          ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   929
      by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   930
  note * = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   931
  show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
   932
    using setsum_joint_distribution_singleton[OF fY fX]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   933
    unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   934
    unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   935
    by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   936
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   937
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39092
diff changeset
   938
section {* Partitioning *}
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
   939
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   940
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   941
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   942
lemma subvimageI:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   943
  assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   944
  shows "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   945
  using assms unfolding subvimage_def by blast
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   946
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   947
lemma subvimageE[consumes 1]:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   948
  assumes "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   949
  obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   950
  using assms unfolding subvimage_def by blast
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   951
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   952
lemma subvimageD:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   953
  "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   954
  using assms unfolding subvimage_def by blast
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   955
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   956
lemma subvimage_subset:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   957
  "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   958
  unfolding subvimage_def by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   959
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   960
lemma subvimage_idem[intro]: "subvimage A g g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   961
  by (safe intro!: subvimageI)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   962
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   963
lemma subvimage_comp_finer[intro]:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   964
  assumes svi: "subvimage A g h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   965
  shows "subvimage A g (f \<circ> h)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   966
proof (rule subvimageI, simp)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   967
  fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   968
  from svi[THEN subvimageD, OF this]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   969
  show "f (h x) = f (h y)" by simp
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   970
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   971
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   972
lemma subvimage_comp_gran:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   973
  assumes svi: "subvimage A g h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   974
  assumes inj: "inj_on f (g ` A)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   975
  shows "subvimage A (f \<circ> g) h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   976
  by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   977
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   978
lemma subvimage_comp:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   979
  assumes svi: "subvimage (f ` A) g h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   980
  shows "subvimage A (g \<circ> f) (h \<circ> f)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   981
  by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   982
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   983
lemma subvimage_trans:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   984
  assumes fg: "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   985
  assumes gh: "subvimage A g h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   986
  shows "subvimage A f h"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   987
  by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   988
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   989
lemma subvimage_translator:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   990
  assumes svi: "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   991
  shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   992
proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   993
  fix x assume "x \<in> A"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   994
  show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   995
    by (rule theI2[of _ "g x"])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   996
      (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   997
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   998
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
   999
lemma subvimage_translator_image:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1000
  assumes svi: "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1001
  shows "\<exists>h. h ` f ` A = g ` A"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1002
proof -
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1003
  from subvimage_translator[OF svi]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1004
  obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1005
  thus ?thesis
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1006
    by (auto intro!: exI[of _ h]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1007
      simp: image_compose[symmetric] comp_def cong: image_cong)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1008
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1009
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1010
lemma subvimage_finite:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1011
  assumes svi: "subvimage A f g" and fin: "finite (f`A)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1012
  shows "finite (g`A)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1013
proof -
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1014
  from subvimage_translator_image[OF svi]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1015
  obtain h where "g`A = h`f`A" by fastsimp
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1016
  with fin show "finite (g`A)" by simp
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1017
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1018
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1019
lemma subvimage_disj:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1020
  assumes svi: "subvimage A f g"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1021
  shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1022
      f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1023
proof (rule disjCI)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1024
  assume "\<not> ?dist"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1025
  then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1026
  thus "?sub" using svi unfolding subvimage_def by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1027
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1028
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1029
lemma setsum_image_split:
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1030
  assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1031
  shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1032
    (is "?lhs = ?rhs")
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1033
proof -
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1034
  have "f ` A =
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1035
      snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1036
      (is "_ = snd ` ?SIGMA")
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1037
    unfolding image_split_eq_Sigma[symmetric]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1038
    by (simp add: image_compose[symmetric] comp_def)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1039
  moreover
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1040
  have snd_inj: "inj_on snd ?SIGMA"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1041
    unfolding image_split_eq_Sigma[symmetric]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1042
    by (auto intro!: inj_onI subvimageD[OF svi])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1043
  ultimately
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1044
  have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1045
    by (auto simp: setsum_reindex intro: setsum_cong)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1046
  also have "... = ?rhs"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1047
    using subvimage_finite[OF svi fin] fin
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1048
    apply (subst setsum_Sigma[symmetric])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1049
    by (auto intro!: finite_subset[of _ "f`A"])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1050
  finally show ?thesis .
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1051
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1052
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1053
lemma (in information_space) entropy_partition:
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
  1054
  assumes sf: "simple_function M X" "simple_function M P"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1055
  assumes svi: "subvimage (space M) X P"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1056
  shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1057
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1058
  let "?XP x p" = "joint_distribution X P {(x, p)}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1059
  let "?X x" = "distribution X {x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1060
  let "?P p" = "distribution P {p}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1061
  note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1062
  note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1063
  note finite_distribution_order[OF fX fP, simp]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1064
  have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1065
    (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1066
  proof (subst setsum_image_split[OF svi],
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1067
      safe intro!: setsum_mono_zero_cong_left imageI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1068
    show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1069
      using sf unfolding simple_function_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1070
  next
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1071
    fix p x assume in_space: "p \<in> space M" "x \<in> space M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1072
    assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1073
    hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1074
    with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1075
    show "x \<in> P -` {P p}" by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1076
  next
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1077
    fix p x assume in_space: "p \<in> space M" "x \<in> space M"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1078
    assume "P x = P p"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1079
    from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1080
    have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1081
      by auto
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1082
    hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1083
      by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1084
    thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1085
      by (auto simp: distribution_def)
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1086
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1087
  moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1088
      ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1089
    by (auto simp add: log_simps zero_less_mult_iff field_simps)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1090
  ultimately show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1091
    unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1092
    using setsum_joint_distribution_singleton[OF fX fP]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41833
diff changeset
  1093
    by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1094
      setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1095
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1096
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1097
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
  1098
  assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1099
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1100
  note X
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1101
  moreover have fX: "simple_function M (f \<circ> X)" using X by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1102
  moreover have "subvimage (space M) X (f \<circ> X)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1103
  ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1104
  then show "\<H>(f \<circ> X) \<le> \<H>(X)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1105
    by (auto intro: conditional_entropy_positive[OF X fX])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1106
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1107
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1108
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
  1109
  assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1110
  shows "\<H>(f \<circ> X) = \<H>(X)"
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1111
proof (rule antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1112
  show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1113
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
  1114
  have sf: "simple_function M (f \<circ> X)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1115
    using X by auto
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1116
  have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1117
    by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1118
  also have "... \<le> \<H>(f \<circ> X)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
  1119
    using entropy_data_processing[OF sf] .
36624
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1120
  finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1121
qed
25153c08655e Cleanup information theory
hoelzl
parents: 36623
diff changeset
  1122
36080
0d9affa4e73c Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff changeset
  1123
end