src/HOL/Probability/Information.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 50419 3177d0374701
child 53015 a1119cf551e8
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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(*  Title:      HOL/Probability/Information.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Information theory*}
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theory Information
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imports
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  Independent_Family
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  Radon_Nikodym
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  "~~/src/HOL/Library/Convex"
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begin
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lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
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  by (subst log_le_cancel_iff) auto
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lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
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  by (subst log_less_cancel_iff) auto
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lemma setsum_cartesian_product':
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  "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
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  unfolding setsum_cartesian_product by simp
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lemma split_pairs:
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  "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
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  "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
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section "Information theory"
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locale information_space = prob_space +
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  fixes b :: real assumes b_gt_1: "1 < b"
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context information_space
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begin
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text {* Introduce some simplification rules for logarithm of base @{term b}. *}
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lemma log_neg_const:
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  assumes "x \<le> 0"
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  shows "log b x = log b 0"
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proof -
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  { fix u :: real
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    have "x \<le> 0" by fact
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    also have "0 < exp u"
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      using exp_gt_zero .
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    finally have "exp u \<noteq> x"
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      by auto }
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  then show "log b x = log b 0"
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    by (simp add: log_def ln_def)
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qed
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lemma log_mult_eq:
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  "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
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  using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
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  by (auto simp: zero_less_mult_iff mult_le_0_iff)
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lemma log_inverse_eq:
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  "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
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  using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
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lemma log_divide_eq:
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  "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
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  unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
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  by (auto simp: zero_less_mult_iff mult_le_0_iff)
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lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
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end
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subsection "Kullback$-$Leibler divergence"
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text {* The Kullback$-$Leibler divergence is also known as relative entropy or
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Kullback$-$Leibler distance. *}
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definition
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  "entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N"
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definition
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  "KL_divergence b M N = integral\<^isup>L N (entropy_density b M N)"
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lemma (in information_space) measurable_entropy_density:
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  assumes ac: "absolutely_continuous M N" "sets N = events"
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  shows "entropy_density b M N \<in> borel_measurable M"
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proof -
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  from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis
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    unfolding entropy_density_def by auto
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qed
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lemma borel_measurable_RN_deriv_density[measurable (raw)]:
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  "f \<in> borel_measurable M \<Longrightarrow> RN_deriv M (density M f) \<in> borel_measurable M"
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  using borel_measurable_RN_deriv_density[of "\<lambda>x. max 0 (f x )" M]
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  by (simp add: density_max_0[symmetric])
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lemma (in sigma_finite_measure) KL_density:
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  fixes f :: "'a \<Rightarrow> real"
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  assumes "1 < b"
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  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
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  shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)"
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  unfolding KL_divergence_def
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proof (subst integral_density)
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  show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
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    using f
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    by (auto simp: comp_def entropy_density_def)
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  have "density M (RN_deriv M (density M f)) = density M f"
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    using f by (intro density_RN_deriv_density) auto
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  then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
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    using f
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    by (intro density_unique)
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       (auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg)
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  show "(\<integral>x. f x * entropy_density b M (density M (\<lambda>x. ereal (f x))) x \<partial>M) = (\<integral>x. f x * log b (f x) \<partial>M)"
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    apply (intro integral_cong_AE)
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    using eq
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    apply eventually_elim
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    apply (auto simp: entropy_density_def)
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    done
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qed fact+
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lemma (in sigma_finite_measure) KL_density_density:
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  fixes f g :: "'a \<Rightarrow> real"
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  assumes "1 < b"
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  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
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  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
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  assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
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  shows "KL_divergence b (density M f) (density M g) = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
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proof -
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  interpret Mf: sigma_finite_measure "density M f"
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    using f by (subst sigma_finite_iff_density_finite) auto
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  have "KL_divergence b (density M f) (density M g) =
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    KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
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    using f g ac by (subst density_density_divide) simp_all
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  also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)"
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    using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density divide_nonneg_nonneg)
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  also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
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    using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE)
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  finally show ?thesis .
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qed
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lemma (in information_space) KL_gt_0:
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  fixes D :: "'a \<Rightarrow> real"
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  assumes "prob_space (density M D)"
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  assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
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  assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
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  assumes A: "density M D \<noteq> M"
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  shows "0 < KL_divergence b M (density M D)"
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proof -
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  interpret N: prob_space "density M D" by fact
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  obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A"
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    using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto
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  let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
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  have [simp, intro]: "?D_set \<in> sets M"
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    using D by auto
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  have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0"
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    using D by (subst positive_integral_0_iff_AE) auto
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  have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)"
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    using D by (simp add: emeasure_density cong: positive_integral_cong)
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  then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1"
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    using N.emeasure_space_1 by simp
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  have "integrable M D" "integral\<^isup>L M D = 1"
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    using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all
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  have "0 \<le> 1 - measure M ?D_set"
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    using prob_le_1 by (auto simp: field_simps)
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  also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
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    using `integrable M D` `integral\<^isup>L M D = 1`
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    by (simp add: emeasure_eq_measure)
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  also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
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  proof (rule integral_less_AE)
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    show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
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      using `integrable M D`
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      by (intro integral_diff integral_indicator) auto
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  next
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    from integral_cmult(1)[OF int, of "ln b"]
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    show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" 
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      by (simp add: ac_simps)
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  next
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    show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
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    proof
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      assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
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      then have disj: "AE x in M. D x = 1 \<or> D x = 0"
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        using D(1) by (auto intro!: AE_I[OF subset_refl] sets.sets_Collect)
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      have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
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        using D(1) by auto
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      also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) \<partial>M)"
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        using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
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      finally have "AE x in M. D x = 1"
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        using D D_pos by (intro AE_I_eq_1) auto
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      then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)"
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        by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
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      also have "\<dots> = density M D A"
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        using `A \<in> sets M` D by (simp add: emeasure_density)
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      finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp
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    qed
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    show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
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      using D(1) by (auto intro: sets.sets_Collect_conj)
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    show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
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      D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))"
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      using D(2)
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    proof (eventually_elim, safe)
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      fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t"
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        and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))"
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      have "D t - 1 = D t - indicator ?D_set t"
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        using Dt by simp
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      also note eq
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      also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
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        using b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
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        by (simp add: log_def ln_div less_le)
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      finally have "ln (1 / D t) = 1 / D t - 1"
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        using `D t \<noteq> 0` by (auto simp: field_simps)
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      from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
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      show False by auto
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    qed
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    show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
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      using D(2) AE_space
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    proof eventually_elim
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      fix t assume "t \<in> space M" "0 \<le> D t"
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      show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
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      proof cases
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        assume asm: "D t \<noteq> 0"
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        then have "0 < D t" using `0 \<le> D t` by auto
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        then have "0 < 1 / D t" by auto
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        have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
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          using asm `t \<in> space M` by (simp add: field_simps)
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        also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
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          using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
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        also have "\<dots> = D t * (ln b * log b (D t))"
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          using `0 < D t` b_gt_1
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          by (simp_all add: log_def ln_div)
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        finally show ?thesis by simp
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      qed simp
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    qed
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  qed
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  also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)"
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    by (simp add: ac_simps)
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  also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)"
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    using int by (rule integral_cmult)
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  finally show ?thesis
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    using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
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qed
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lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0"
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proof -
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  have "AE x in M. 1 = RN_deriv M M x"
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  proof (rule RN_deriv_unique)
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    show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto
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    show "density M (\<lambda>x. 1) = M"
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      apply (auto intro!: measure_eqI emeasure_density)
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      apply (subst emeasure_density)
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      apply auto
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      done
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  qed
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  then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
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    by (elim AE_mp) simp
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  from integral_cong_AE[OF this]
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  have "integral\<^isup>L M (entropy_density b M M) = 0"
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    by (simp add: entropy_density_def comp_def)
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  then show "KL_divergence b M M = 0"
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    unfolding KL_divergence_def
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    by auto
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qed
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lemma (in information_space) KL_eq_0_iff_eq:
hoelzl@47694
   272
  fixes D :: "'a \<Rightarrow> real"
hoelzl@47694
   273
  assumes "prob_space (density M D)"
hoelzl@47694
   274
  assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
hoelzl@47694
   275
  assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694
   276
  shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M"
hoelzl@47694
   277
  using KL_same_eq_0[of b] KL_gt_0[OF assms]
hoelzl@47694
   278
  by (auto simp: less_le)
hoelzl@43340
   279
hoelzl@47694
   280
lemma (in information_space) KL_eq_0_iff_eq_ac:
hoelzl@47694
   281
  fixes D :: "'a \<Rightarrow> real"
hoelzl@47694
   282
  assumes "prob_space N"
hoelzl@47694
   283
  assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694
   284
  assumes int: "integrable N (entropy_density b M N)"
hoelzl@47694
   285
  shows "KL_divergence b M N = 0 \<longleftrightarrow> N = M"
hoelzl@41833
   286
proof -
hoelzl@47694
   287
  interpret N: prob_space N by fact
hoelzl@47694
   288
  have "finite_measure N" by unfold_locales
hoelzl@47694
   289
  from real_RN_deriv[OF this ac] guess D . note D = this
hoelzl@47694
   290
  
hoelzl@47694
   291
  have "N = density M (RN_deriv M N)"
hoelzl@47694
   292
    using ac by (rule density_RN_deriv[symmetric])
hoelzl@47694
   293
  also have "\<dots> = density M D"
hoelzl@47694
   294
    using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong)
hoelzl@47694
   295
  finally have N: "N = density M D" .
hoelzl@41833
   296
hoelzl@47694
   297
  from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
hoelzl@47694
   298
  have "integrable N (\<lambda>x. log b (D x))"
hoelzl@47694
   299
    by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
hoelzl@47694
   300
       (auto simp: N entropy_density_def)
hoelzl@47694
   301
  with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694
   302
    by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def)
hoelzl@47694
   303
  with `prob_space N` D show ?thesis
hoelzl@47694
   304
    unfolding N
hoelzl@47694
   305
    by (intro KL_eq_0_iff_eq) auto
hoelzl@41833
   306
qed
hoelzl@41833
   307
hoelzl@47694
   308
lemma (in information_space) KL_nonneg:
hoelzl@47694
   309
  assumes "prob_space (density M D)"
hoelzl@47694
   310
  assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
hoelzl@47694
   311
  assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694
   312
  shows "0 \<le> KL_divergence b M (density M D)"
hoelzl@47694
   313
  using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0)
hoelzl@40859
   314
hoelzl@47694
   315
lemma (in sigma_finite_measure) KL_density_density_nonneg:
hoelzl@47694
   316
  fixes f g :: "'a \<Rightarrow> real"
hoelzl@47694
   317
  assumes "1 < b"
hoelzl@47694
   318
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)"
hoelzl@47694
   319
  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)"
hoelzl@47694
   320
  assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
hoelzl@47694
   321
  assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))"
hoelzl@47694
   322
  shows "0 \<le> KL_divergence b (density M f) (density M g)"
hoelzl@47694
   323
proof -
hoelzl@47694
   324
  interpret Mf: prob_space "density M f" by fact
hoelzl@47694
   325
  interpret Mf: information_space "density M f" b by default fact
hoelzl@47694
   326
  have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _")
hoelzl@47694
   327
    using f g ac by (subst density_density_divide) simp_all
hoelzl@36080
   328
hoelzl@47694
   329
  have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
hoelzl@47694
   330
  proof (rule Mf.KL_nonneg)
hoelzl@47694
   331
    show "prob_space ?DD" unfolding eq by fact
hoelzl@47694
   332
    from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)"
hoelzl@47694
   333
      by auto
hoelzl@47694
   334
    show "AE x in density M f. 0 \<le> g x / f x"
hoelzl@47694
   335
      using f g by (auto simp: AE_density divide_nonneg_nonneg)
hoelzl@47694
   336
    show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
hoelzl@47694
   337
      using `1 < b` f g ac
hoelzl@47694
   338
      by (subst integral_density)
hoelzl@47694
   339
         (auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
hoelzl@47694
   340
  qed
hoelzl@47694
   341
  also have "\<dots> = KL_divergence b (density M f) (density M g)"
hoelzl@47694
   342
    using f g ac by (subst density_density_divide) simp_all
hoelzl@47694
   343
  finally show ?thesis .
hoelzl@36080
   344
qed
hoelzl@36080
   345
hoelzl@49803
   346
subsection {* Finite Entropy *}
hoelzl@49803
   347
hoelzl@49803
   348
definition (in information_space) 
hoelzl@49803
   349
  "finite_entropy S X f \<longleftrightarrow> distributed M S X f \<and> integrable S (\<lambda>x. f x * log b (f x))"
hoelzl@49803
   350
hoelzl@49803
   351
lemma (in information_space) finite_entropy_simple_function:
hoelzl@49803
   352
  assumes X: "simple_function M X"
hoelzl@49803
   353
  shows "finite_entropy (count_space (X`space M)) X (\<lambda>a. measure M {x \<in> space M. X x = a})"
hoelzl@49803
   354
  unfolding finite_entropy_def
hoelzl@49803
   355
proof
hoelzl@49803
   356
  have [simp]: "finite (X ` space M)"
hoelzl@49803
   357
    using X by (auto simp: simple_function_def)
hoelzl@49803
   358
  then show "integrable (count_space (X ` space M))
hoelzl@49803
   359
     (\<lambda>x. prob {xa \<in> space M. X xa = x} * log b (prob {xa \<in> space M. X xa = x}))"
hoelzl@49803
   360
    by (rule integrable_count_space)
hoelzl@49803
   361
  have d: "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then prob {xa \<in> space M. X xa = x} else 0))"
hoelzl@49803
   362
    by (rule distributed_simple_function_superset[OF X]) (auto intro!: arg_cong[where f=prob])
hoelzl@49803
   363
  show "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (prob {xa \<in> space M. X xa = x}))"
hoelzl@49803
   364
    by (rule distributed_cong_density[THEN iffD1, OF _ _ _ d]) auto
hoelzl@49803
   365
qed
hoelzl@49803
   366
hoelzl@49803
   367
lemma distributed_transform_AE:
hoelzl@49803
   368
  assumes T: "T \<in> measurable P Q" "absolutely_continuous Q (distr P Q T)"
hoelzl@49803
   369
  assumes g: "distributed M Q Y g"
hoelzl@49803
   370
  shows "AE x in P. 0 \<le> g (T x)"
hoelzl@49803
   371
  using g
hoelzl@49803
   372
  apply (subst AE_distr_iff[symmetric, OF T(1)])
hoelzl@50003
   373
  apply simp
hoelzl@49803
   374
  apply (rule absolutely_continuous_AE[OF _ T(2)])
hoelzl@49803
   375
  apply simp
hoelzl@49803
   376
  apply (simp add: distributed_AE)
hoelzl@49803
   377
  done
hoelzl@49803
   378
hoelzl@49803
   379
lemma ac_fst:
hoelzl@49803
   380
  assumes "sigma_finite_measure T"
hoelzl@49803
   381
  shows "absolutely_continuous S (distr (S \<Otimes>\<^isub>M T) S fst)"
hoelzl@49803
   382
proof -
hoelzl@49803
   383
  interpret sigma_finite_measure T by fact
hoelzl@49803
   384
  { fix A assume "A \<in> sets S" "emeasure S A = 0"
hoelzl@49803
   385
    moreover then have "fst -` A \<inter> space (S \<Otimes>\<^isub>M T) = A \<times> space T"
immler@50244
   386
      by (auto simp: space_pair_measure dest!: sets.sets_into_space)
hoelzl@49803
   387
    ultimately have "emeasure (S \<Otimes>\<^isub>M T) (fst -` A \<inter> space (S \<Otimes>\<^isub>M T)) = 0"
hoelzl@49803
   388
      by (simp add: emeasure_pair_measure_Times) }
hoelzl@49803
   389
  then show ?thesis
hoelzl@49803
   390
    unfolding absolutely_continuous_def
hoelzl@49803
   391
    apply (auto simp: null_sets_distr_iff)
hoelzl@49803
   392
    apply (auto simp: null_sets_def intro!: measurable_sets)
hoelzl@49803
   393
    done
hoelzl@49803
   394
qed
hoelzl@49803
   395
hoelzl@49803
   396
lemma ac_snd:
hoelzl@49803
   397
  assumes "sigma_finite_measure T"
hoelzl@49803
   398
  shows "absolutely_continuous T (distr (S \<Otimes>\<^isub>M T) T snd)"
hoelzl@49803
   399
proof -
hoelzl@49803
   400
  interpret sigma_finite_measure T by fact
hoelzl@49803
   401
  { fix A assume "A \<in> sets T" "emeasure T A = 0"
hoelzl@49803
   402
    moreover then have "snd -` A \<inter> space (S \<Otimes>\<^isub>M T) = space S \<times> A"
immler@50244
   403
      by (auto simp: space_pair_measure dest!: sets.sets_into_space)
hoelzl@49803
   404
    ultimately have "emeasure (S \<Otimes>\<^isub>M T) (snd -` A \<inter> space (S \<Otimes>\<^isub>M T)) = 0"
hoelzl@49803
   405
      by (simp add: emeasure_pair_measure_Times) }
hoelzl@49803
   406
  then show ?thesis
hoelzl@49803
   407
    unfolding absolutely_continuous_def
hoelzl@49803
   408
    apply (auto simp: null_sets_distr_iff)
hoelzl@49803
   409
    apply (auto simp: null_sets_def intro!: measurable_sets)
hoelzl@49803
   410
    done
hoelzl@49803
   411
qed
hoelzl@49803
   412
hoelzl@49803
   413
lemma distributed_integrable:
hoelzl@49803
   414
  "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow>
hoelzl@49803
   415
    integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
hoelzl@50003
   416
  by (auto simp: distributed_real_AE
hoelzl@49803
   417
                    distributed_distr_eq_density[symmetric] integral_density[symmetric] integrable_distr_eq)
hoelzl@49803
   418
  
hoelzl@49803
   419
lemma distributed_transform_integrable:
hoelzl@49803
   420
  assumes Px: "distributed M N X Px"
hoelzl@49803
   421
  assumes "distributed M P Y Py"
hoelzl@49803
   422
  assumes Y: "Y = (\<lambda>x. T (X x))" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
hoelzl@49803
   423
  shows "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@49803
   424
proof -
hoelzl@49803
   425
  have "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable M (\<lambda>x. f (Y x))"
hoelzl@49803
   426
    by (rule distributed_integrable) fact+
hoelzl@49803
   427
  also have "\<dots> \<longleftrightarrow> integrable M (\<lambda>x. f (T (X x)))"
hoelzl@49803
   428
    using Y by simp
hoelzl@49803
   429
  also have "\<dots> \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@49803
   430
    using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
hoelzl@49803
   431
  finally show ?thesis .
hoelzl@49803
   432
qed
hoelzl@49803
   433
hoelzl@49803
   434
lemma integrable_cong_AE_imp: "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
hoelzl@49803
   435
  using integrable_cong_AE by blast
hoelzl@49803
   436
hoelzl@49803
   437
lemma (in information_space) finite_entropy_integrable:
hoelzl@49803
   438
  "finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))"
hoelzl@49803
   439
  unfolding finite_entropy_def by auto
hoelzl@49803
   440
hoelzl@49803
   441
lemma (in information_space) finite_entropy_distributed:
hoelzl@49803
   442
  "finite_entropy S X Px \<Longrightarrow> distributed M S X Px"
hoelzl@49803
   443
  unfolding finite_entropy_def by auto
hoelzl@49803
   444
hoelzl@49803
   445
lemma (in information_space) finite_entropy_integrable_transform:
hoelzl@49803
   446
  assumes Fx: "finite_entropy S X Px"
hoelzl@49803
   447
  assumes Fy: "distributed M T Y Py"
hoelzl@49803
   448
    and "X = (\<lambda>x. f (Y x))"
hoelzl@49803
   449
    and "f \<in> measurable T S"
hoelzl@49803
   450
  shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))"
hoelzl@49803
   451
  using assms unfolding finite_entropy_def
hoelzl@49803
   452
  using distributed_transform_integrable[of M T Y Py S X Px f "\<lambda>x. log b (Px x)"]
hoelzl@50003
   453
  by auto
hoelzl@49803
   454
hoelzl@39097
   455
subsection {* Mutual Information *}
hoelzl@39097
   456
hoelzl@36080
   457
definition (in prob_space)
hoelzl@38656
   458
  "mutual_information b S T X Y =
hoelzl@47694
   459
    KL_divergence b (distr M S X \<Otimes>\<^isub>M distr M T Y) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))"
hoelzl@36080
   460
hoelzl@47694
   461
lemma (in information_space) mutual_information_indep_vars:
hoelzl@43340
   462
  fixes S T X Y
hoelzl@47694
   463
  defines "P \<equiv> distr M S X \<Otimes>\<^isub>M distr M T Y"
hoelzl@47694
   464
  defines "Q \<equiv> distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@43340
   465
  shows "indep_var S X T Y \<longleftrightarrow>
hoelzl@43340
   466
    (random_variable S X \<and> random_variable T Y \<and>
hoelzl@47694
   467
      absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and>
hoelzl@47694
   468
      mutual_information b S T X Y = 0)"
hoelzl@47694
   469
  unfolding indep_var_distribution_eq
hoelzl@43340
   470
proof safe
hoelzl@50003
   471
  assume rv[measurable]: "random_variable S X" "random_variable T Y"
hoelzl@43340
   472
hoelzl@47694
   473
  interpret X: prob_space "distr M S X"
hoelzl@47694
   474
    by (rule prob_space_distr) fact
hoelzl@47694
   475
  interpret Y: prob_space "distr M T Y"
hoelzl@47694
   476
    by (rule prob_space_distr) fact
hoelzl@47694
   477
  interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default
hoelzl@47694
   478
  interpret P: information_space P b unfolding P_def by default (rule b_gt_1)
hoelzl@43340
   479
hoelzl@47694
   480
  interpret Q: prob_space Q unfolding Q_def
hoelzl@50003
   481
    by (rule prob_space_distr) simp
hoelzl@43340
   482
hoelzl@47694
   483
  { assume "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694
   484
    then have [simp]: "Q = P"  unfolding Q_def P_def by simp
hoelzl@43340
   485
hoelzl@47694
   486
    show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
hoelzl@47694
   487
    then have ed: "entropy_density b P Q \<in> borel_measurable P"
hoelzl@47694
   488
      by (rule P.measurable_entropy_density) simp
hoelzl@43340
   489
hoelzl@47694
   490
    have "AE x in P. 1 = RN_deriv P Q x"
hoelzl@47694
   491
    proof (rule P.RN_deriv_unique)
hoelzl@47694
   492
      show "density P (\<lambda>x. 1) = Q"
hoelzl@47694
   493
        unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density)
hoelzl@47694
   494
    qed auto
hoelzl@47694
   495
    then have ae_0: "AE x in P. entropy_density b P Q x = 0"
hoelzl@47694
   496
      by eventually_elim (auto simp: entropy_density_def)
hoelzl@47694
   497
    then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)"
hoelzl@47694
   498
      using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
hoelzl@47694
   499
    then show "integrable Q (entropy_density b P Q)" by simp
hoelzl@43340
   500
hoelzl@47694
   501
    show "mutual_information b S T X Y = 0"
hoelzl@47694
   502
      unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
hoelzl@47694
   503
      using ae_0 by (simp cong: integral_cong_AE) }
hoelzl@43340
   504
hoelzl@47694
   505
  { assume ac: "absolutely_continuous P Q"
hoelzl@47694
   506
    assume int: "integrable Q (entropy_density b P Q)"
hoelzl@47694
   507
    assume I_eq_0: "mutual_information b S T X Y = 0"
hoelzl@43340
   508
hoelzl@47694
   509
    have eq: "Q = P"
hoelzl@47694
   510
    proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1])
hoelzl@47694
   511
      show "prob_space Q" by unfold_locales
hoelzl@47694
   512
      show "absolutely_continuous P Q" by fact
hoelzl@47694
   513
      show "integrable Q (entropy_density b P Q)" by fact
hoelzl@47694
   514
      show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure)
hoelzl@47694
   515
      show "KL_divergence b P Q = 0"
hoelzl@47694
   516
        using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def)
hoelzl@47694
   517
    qed
hoelzl@47694
   518
    then show "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694
   519
      unfolding P_def Q_def .. }
hoelzl@43340
   520
qed
hoelzl@43340
   521
hoelzl@40859
   522
abbreviation (in information_space)
hoelzl@40859
   523
  mutual_information_Pow ("\<I>'(_ ; _')") where
hoelzl@47694
   524
  "\<I>(X ; Y) \<equiv> mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y"
hoelzl@41689
   525
hoelzl@47694
   526
lemma (in information_space)
hoelzl@47694
   527
  fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@49803
   528
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49803
   529
  assumes Fx: "finite_entropy S X Px" and Fy: "finite_entropy T Y Py"
hoelzl@49803
   530
  assumes Fxy: "finite_entropy (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803
   531
  defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@49803
   532
  shows mutual_information_distr': "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R")
hoelzl@49803
   533
    and mutual_information_nonneg': "0 \<le> mutual_information b S T X Y"
hoelzl@49803
   534
proof -
hoelzl@49803
   535
  have Px: "distributed M S X Px"
hoelzl@49803
   536
    using Fx by (auto simp: finite_entropy_def)
hoelzl@49803
   537
  have Py: "distributed M T Y Py"
hoelzl@49803
   538
    using Fy by (auto simp: finite_entropy_def)
hoelzl@49803
   539
  have Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803
   540
    using Fxy by (auto simp: finite_entropy_def)
hoelzl@49803
   541
hoelzl@49803
   542
  have X: "random_variable S X"
hoelzl@50003
   543
    using Px by auto
hoelzl@49803
   544
  have Y: "random_variable T Y"
hoelzl@50003
   545
    using Py by auto
hoelzl@49803
   546
  interpret S: sigma_finite_measure S by fact
hoelzl@49803
   547
  interpret T: sigma_finite_measure T by fact
hoelzl@49803
   548
  interpret ST: pair_sigma_finite S T ..
hoelzl@49803
   549
  interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
hoelzl@49803
   550
  interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
hoelzl@49803
   551
  interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
hoelzl@49803
   552
  let ?P = "S \<Otimes>\<^isub>M T"
hoelzl@49803
   553
  let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"
hoelzl@49803
   554
hoelzl@49803
   555
  { fix A assume "A \<in> sets S"
hoelzl@49803
   556
    with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
hoelzl@49803
   557
      by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@49803
   558
               intro!: arg_cong[where f="emeasure M"]) }
hoelzl@49803
   559
  note marginal_eq1 = this
hoelzl@49803
   560
  { fix A assume "A \<in> sets T"
hoelzl@49803
   561
    with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
hoelzl@49803
   562
      by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@49803
   563
               intro!: arg_cong[where f="emeasure M"]) }
hoelzl@49803
   564
  note marginal_eq2 = this
hoelzl@49803
   565
hoelzl@49803
   566
  have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
hoelzl@49803
   567
    by auto
hoelzl@49803
   568
hoelzl@49803
   569
  have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
hoelzl@49803
   570
    unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
hoelzl@49803
   571
  proof (subst pair_measure_density)
hoelzl@49803
   572
    show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
hoelzl@49803
   573
      "AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
hoelzl@49803
   574
      using Px Py by (auto simp: distributed_def)
hoelzl@49803
   575
    show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
hoelzl@49803
   576
  qed (fact | simp)+
hoelzl@49803
   577
  
hoelzl@49803
   578
  have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))"
hoelzl@49803
   579
    unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..
hoelzl@49803
   580
hoelzl@49803
   581
  from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
hoelzl@49803
   582
    by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
hoelzl@49803
   583
  have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
hoelzl@49803
   584
  proof (rule ST.AE_pair_measure)
hoelzl@49803
   585
    show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
hoelzl@49803
   586
      using f by auto
hoelzl@49803
   587
    show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
hoelzl@49803
   588
      using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
hoelzl@49803
   589
  qed
hoelzl@49803
   590
hoelzl@49803
   591
  have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@49803
   592
    by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@49803
   593
  moreover
hoelzl@49803
   594
  have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@49803
   595
    by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@49803
   596
  ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@49803
   597
    by eventually_elim auto
hoelzl@49803
   598
hoelzl@49803
   599
  show "?M = ?R"
hoelzl@49803
   600
    unfolding M f_def
hoelzl@49803
   601
    using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
hoelzl@49803
   602
    by (rule ST.KL_density_density)
hoelzl@49803
   603
hoelzl@49803
   604
  have X: "X = fst \<circ> (\<lambda>x. (X x, Y x))" and Y: "Y = snd \<circ> (\<lambda>x. (X x, Y x))"
hoelzl@49803
   605
    by auto
hoelzl@49803
   606
hoelzl@49803
   607
  have "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)))"
hoelzl@49803
   608
    using finite_entropy_integrable[OF Fxy]
hoelzl@49803
   609
    using finite_entropy_integrable_transform[OF Fx Pxy, of fst]
hoelzl@49803
   610
    using finite_entropy_integrable_transform[OF Fy Pxy, of snd]
hoelzl@49803
   611
    by simp
hoelzl@49803
   612
  moreover have "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)"
hoelzl@49803
   613
    unfolding f_def using Px Py Pxy
hoelzl@49803
   614
    by (auto intro: distributed_real_measurable measurable_fst'' measurable_snd''
hoelzl@49803
   615
      intro!: borel_measurable_times borel_measurable_log borel_measurable_divide)
hoelzl@49803
   616
  ultimately have int: "integrable (S \<Otimes>\<^isub>M T) f"
hoelzl@49803
   617
    apply (rule integrable_cong_AE_imp)
hoelzl@49803
   618
    using
hoelzl@49803
   619
      distributed_transform_AE[OF measurable_fst ac_fst, of T, OF T Px]
hoelzl@49803
   620
      distributed_transform_AE[OF measurable_snd ac_snd, of _ _ _ _ S, OF T Py]
hoelzl@49803
   621
      subdensity_real[OF measurable_fst Pxy Px X]
hoelzl@49803
   622
      subdensity_real[OF measurable_snd Pxy Py Y]
hoelzl@49803
   623
      distributed_real_AE[OF Pxy]
hoelzl@49803
   624
    by eventually_elim
hoelzl@49803
   625
       (auto simp: f_def log_divide_eq log_mult_eq field_simps zero_less_mult_iff mult_nonneg_nonneg)
hoelzl@49803
   626
hoelzl@49803
   627
  show "0 \<le> ?M" unfolding M
hoelzl@49803
   628
  proof (rule ST.KL_density_density_nonneg
hoelzl@49803
   629
    [OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
hoelzl@49803
   630
    show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) "
hoelzl@49803
   631
      unfolding distributed_distr_eq_density[OF Pxy, symmetric]
hoelzl@49803
   632
      using distributed_measurable[OF Pxy] by (rule prob_space_distr)
hoelzl@49803
   633
    show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
hoelzl@49803
   634
      unfolding distr_eq[symmetric] by unfold_locales
hoelzl@49803
   635
  qed
hoelzl@49803
   636
qed
hoelzl@49803
   637
hoelzl@49803
   638
hoelzl@49803
   639
lemma (in information_space)
hoelzl@49803
   640
  fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694
   641
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@47694
   642
  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
hoelzl@47694
   643
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
   644
  defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@47694
   645
  shows mutual_information_distr: "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R")
hoelzl@47694
   646
    and mutual_information_nonneg: "integrable (S \<Otimes>\<^isub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y"
hoelzl@40859
   647
proof -
hoelzl@47694
   648
  have X: "random_variable S X"
hoelzl@47694
   649
    using Px by (auto simp: distributed_def)
hoelzl@47694
   650
  have Y: "random_variable T Y"
hoelzl@47694
   651
    using Py by (auto simp: distributed_def)
hoelzl@47694
   652
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
   653
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
   654
  interpret ST: pair_sigma_finite S T ..
hoelzl@47694
   655
  interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
hoelzl@47694
   656
  interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
hoelzl@47694
   657
  interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
hoelzl@47694
   658
  let ?P = "S \<Otimes>\<^isub>M T"
hoelzl@47694
   659
  let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"
hoelzl@47694
   660
hoelzl@47694
   661
  { fix A assume "A \<in> sets S"
hoelzl@47694
   662
    with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
hoelzl@47694
   663
      by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@47694
   664
               intro!: arg_cong[where f="emeasure M"]) }
hoelzl@47694
   665
  note marginal_eq1 = this
hoelzl@47694
   666
  { fix A assume "A \<in> sets T"
hoelzl@47694
   667
    with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
hoelzl@47694
   668
      by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@47694
   669
               intro!: arg_cong[where f="emeasure M"]) }
hoelzl@47694
   670
  note marginal_eq2 = this
hoelzl@47694
   671
hoelzl@47694
   672
  have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
hoelzl@47694
   673
    by auto
hoelzl@47694
   674
hoelzl@47694
   675
  have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
hoelzl@47694
   676
    unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
hoelzl@47694
   677
  proof (subst pair_measure_density)
hoelzl@47694
   678
    show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
hoelzl@47694
   679
      "AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
hoelzl@47694
   680
      using Px Py by (auto simp: distributed_def)
hoelzl@47694
   681
    show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
hoelzl@47694
   682
  qed (fact | simp)+
hoelzl@47694
   683
  
hoelzl@47694
   684
  have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))"
hoelzl@47694
   685
    unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..
hoelzl@47694
   686
hoelzl@47694
   687
  from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
hoelzl@47694
   688
    by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
hoelzl@47694
   689
  have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
hoelzl@47694
   690
  proof (rule ST.AE_pair_measure)
hoelzl@47694
   691
    show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
hoelzl@47694
   692
      using f by auto
hoelzl@47694
   693
    show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
hoelzl@47694
   694
      using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
hoelzl@47694
   695
  qed
hoelzl@47694
   696
hoelzl@47694
   697
  have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@47694
   698
    by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@47694
   699
  moreover
hoelzl@47694
   700
  have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@47694
   701
    by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@47694
   702
  ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
   703
    by eventually_elim auto
hoelzl@47694
   704
hoelzl@47694
   705
  show "?M = ?R"
hoelzl@47694
   706
    unfolding M f_def
hoelzl@47694
   707
    using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
hoelzl@47694
   708
    by (rule ST.KL_density_density)
hoelzl@47694
   709
hoelzl@47694
   710
  assume int: "integrable (S \<Otimes>\<^isub>M T) f"
hoelzl@47694
   711
  show "0 \<le> ?M" unfolding M
hoelzl@47694
   712
  proof (rule ST.KL_density_density_nonneg
hoelzl@47694
   713
    [OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
hoelzl@47694
   714
    show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) "
hoelzl@47694
   715
      unfolding distributed_distr_eq_density[OF Pxy, symmetric]
hoelzl@47694
   716
      using distributed_measurable[OF Pxy] by (rule prob_space_distr)
hoelzl@47694
   717
    show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
hoelzl@47694
   718
      unfolding distr_eq[symmetric] by unfold_locales
hoelzl@40859
   719
  qed
hoelzl@40859
   720
qed
hoelzl@40859
   721
hoelzl@40859
   722
lemma (in information_space)
hoelzl@47694
   723
  fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694
   724
  assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@47694
   725
  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
hoelzl@47694
   726
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
   727
  assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
hoelzl@47694
   728
  shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
hoelzl@36624
   729
proof -
hoelzl@47694
   730
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
   731
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
   732
  interpret ST: pair_sigma_finite S T ..
hoelzl@36080
   733
hoelzl@47694
   734
  have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
   735
    by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@47694
   736
  moreover
hoelzl@47694
   737
  have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
   738
    by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@47694
   739
  moreover 
hoelzl@47694
   740
  have "AE x in S \<Otimes>\<^isub>M T. Pxy x = Px (fst x) * Py (snd x)"
hoelzl@47694
   741
    using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy]
hoelzl@47694
   742
    by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'')
hoelzl@47694
   743
  ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
hoelzl@47694
   744
    by eventually_elim simp
hoelzl@47694
   745
  then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@47694
   746
    by (rule integral_cong_AE)
hoelzl@47694
   747
  then show ?thesis
hoelzl@47694
   748
    by (subst mutual_information_distr[OF assms(1-5)]) simp
hoelzl@36080
   749
qed
hoelzl@36080
   750
hoelzl@47694
   751
lemma (in information_space) mutual_information_simple_distributed:
hoelzl@47694
   752
  assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py"
hoelzl@47694
   753
  assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
   754
  shows "\<I>(X ; Y) = (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x))`space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
hoelzl@47694
   755
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@47694
   756
  note fin = simple_distributed_joint_finite[OF XY, simp]
hoelzl@47694
   757
  show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694
   758
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
   759
  show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694
   760
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
   761
  let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
hoelzl@47694
   762
  let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@47694
   763
  have "\<And>x. ?f x = (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) else 0)"
hoelzl@47694
   764
    by auto
hoelzl@47694
   765
  with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M))) =
hoelzl@47694
   766
    (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
hoelzl@47694
   767
    by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta'
hoelzl@47694
   768
             intro!: setsum_cong)
hoelzl@47694
   769
qed
hoelzl@36080
   770
hoelzl@47694
   771
lemma (in information_space)
hoelzl@47694
   772
  fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694
   773
  assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py"
hoelzl@47694
   774
  assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
   775
  assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
hoelzl@47694
   776
  shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0"
hoelzl@47694
   777
proof (subst mutual_information_simple_distributed[OF Px Py Pxy])
hoelzl@47694
   778
  have "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) =
hoelzl@47694
   779
    (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)"
hoelzl@47694
   780
    by (intro setsum_cong) (auto simp: ae)
hoelzl@47694
   781
  then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M.
hoelzl@47694
   782
    Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
hoelzl@47694
   783
qed
hoelzl@36080
   784
hoelzl@39097
   785
subsection {* Entropy *}
hoelzl@39097
   786
hoelzl@47694
   787
definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where
hoelzl@47694
   788
  "entropy b S X = - KL_divergence b S (distr M S X)"
hoelzl@47694
   789
hoelzl@40859
   790
abbreviation (in information_space)
hoelzl@40859
   791
  entropy_Pow ("\<H>'(_')") where
hoelzl@47694
   792
  "\<H>(X) \<equiv> entropy b (count_space (X`space M)) X"
hoelzl@41981
   793
hoelzl@49791
   794
lemma (in prob_space) distributed_RN_deriv:
hoelzl@49791
   795
  assumes X: "distributed M S X Px"
hoelzl@49791
   796
  shows "AE x in S. RN_deriv S (density S Px) x = Px x"
hoelzl@49791
   797
proof -
hoelzl@49791
   798
  note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49791
   799
  interpret X: prob_space "distr M S X"
hoelzl@49791
   800
    using D(1) by (rule prob_space_distr)
hoelzl@49791
   801
hoelzl@49791
   802
  have sf: "sigma_finite_measure (distr M S X)" by default
hoelzl@49791
   803
  show ?thesis
hoelzl@49791
   804
    using D
hoelzl@49791
   805
    apply (subst eq_commute)
hoelzl@49791
   806
    apply (intro RN_deriv_unique_sigma_finite)
hoelzl@49791
   807
    apply (auto intro: divide_nonneg_nonneg measure_nonneg
hoelzl@49791
   808
             simp: distributed_distr_eq_density[symmetric, OF X] sf)
hoelzl@49791
   809
    done
hoelzl@49791
   810
qed
hoelzl@49791
   811
hoelzl@49788
   812
lemma (in information_space)
hoelzl@47694
   813
  fixes X :: "'a \<Rightarrow> 'b"
hoelzl@49785
   814
  assumes X: "distributed M MX X f"
hoelzl@49788
   815
  shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)
hoelzl@49788
   816
proof -
hoelzl@49785
   817
  note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49791
   818
  note ae = distributed_RN_deriv[OF X]
hoelzl@49788
   819
hoelzl@49788
   820
  have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) =
hoelzl@49785
   821
    log b (f x)"
hoelzl@49785
   822
    unfolding distributed_distr_eq_density[OF X]
hoelzl@49785
   823
    apply (subst AE_density)
hoelzl@49785
   824
    using D apply simp
hoelzl@49785
   825
    using ae apply eventually_elim
hoelzl@49785
   826
    apply auto
hoelzl@49785
   827
    done
hoelzl@49788
   828
hoelzl@49788
   829
  have int_eq: "- (\<integral> x. log b (f x) \<partial>distr M MX X) = - (\<integral> x. f x * log b (f x) \<partial>MX)"
hoelzl@49785
   830
    unfolding distributed_distr_eq_density[OF X]
hoelzl@49785
   831
    using D
hoelzl@49785
   832
    by (subst integral_density)
hoelzl@49785
   833
       (auto simp: borel_measurable_ereal_iff)
hoelzl@49788
   834
hoelzl@49788
   835
  show ?eq
hoelzl@49788
   836
    unfolding entropy_def KL_divergence_def entropy_density_def comp_def
hoelzl@49788
   837
    apply (subst integral_cong_AE)
hoelzl@49788
   838
    apply (rule ae_eq)
hoelzl@49788
   839
    apply (rule int_eq)
hoelzl@49788
   840
    done
hoelzl@49788
   841
qed
hoelzl@49785
   842
hoelzl@49786
   843
lemma (in prob_space) distributed_imp_emeasure_nonzero:
hoelzl@49786
   844
  assumes X: "distributed M MX X Px"
hoelzl@49786
   845
  shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0"
hoelzl@49786
   846
proof
hoelzl@49786
   847
  note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786
   848
  interpret X: prob_space "distr M MX X"
hoelzl@49786
   849
    using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@49786
   850
hoelzl@49786
   851
  assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0"
hoelzl@49786
   852
  with Px have "AE x in MX. Px x = 0"
hoelzl@49786
   853
    by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ereal_iff)
hoelzl@49786
   854
  moreover
hoelzl@49786
   855
  from X.emeasure_space_1 have "(\<integral>\<^isup>+x. Px x \<partial>MX) = 1"
hoelzl@49786
   856
    unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786
   857
    by (subst (asm) emeasure_density)
hoelzl@49786
   858
       (auto simp: borel_measurable_ereal_iff intro!: integral_cong cong: positive_integral_cong)
hoelzl@49786
   859
  ultimately show False
hoelzl@49786
   860
    by (simp add: positive_integral_cong_AE)
hoelzl@49786
   861
qed
hoelzl@49786
   862
hoelzl@49786
   863
lemma (in information_space) entropy_le:
hoelzl@49786
   864
  fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
hoelzl@49786
   865
  assumes X: "distributed M MX X Px"
hoelzl@49786
   866
  and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>"
hoelzl@49786
   867
  and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786
   868
  shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
hoelzl@49786
   869
proof -
hoelzl@49786
   870
  note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786
   871
  interpret X: prob_space "distr M MX X"
hoelzl@49786
   872
    using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@49786
   873
hoelzl@49786
   874
  have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) = 
hoelzl@49786
   875
    - log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)"
hoelzl@49786
   876
    using Px fin
hoelzl@49786
   877
    by (subst integral_indicator) (auto simp: measure_def borel_measurable_ereal_iff)
hoelzl@49786
   878
  also have "- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX) = - log b (\<integral> x. 1 / Px x \<partial>distr M MX X)"
hoelzl@49786
   879
    unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786
   880
    apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus])
hoelzl@49786
   881
    by (subst integral_density) (auto simp: borel_measurable_ereal_iff intro!: integral_cong)
hoelzl@49786
   882
  also have "\<dots> \<le> (\<integral> x. - log b (1 / Px x) \<partial>distr M MX X)"
hoelzl@49786
   883
  proof (rule X.jensens_inequality[of "\<lambda>x. 1 / Px x" "{0<..}" 0 1 "\<lambda>x. - log b x"])
hoelzl@49786
   884
    show "AE x in distr M MX X. 1 / Px x \<in> {0<..}"
hoelzl@49786
   885
      unfolding distributed_distr_eq_density[OF X]
hoelzl@49786
   886
      using Px by (auto simp: AE_density)
hoelzl@49786
   887
    have [simp]: "\<And>x. x \<in> space MX \<Longrightarrow> ereal (if Px x = 0 then 0 else 1) = indicator {x \<in> space MX. Px x \<noteq> 0} x"
hoelzl@49786
   888
      by (auto simp: one_ereal_def)
hoelzl@49786
   889
    have "(\<integral>\<^isup>+ x. max 0 (ereal (- (if Px x = 0 then 0 else 1))) \<partial>MX) = (\<integral>\<^isup>+ x. 0 \<partial>MX)"
hoelzl@49786
   890
      by (intro positive_integral_cong) (auto split: split_max)
hoelzl@49786
   891
    then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)"
hoelzl@49786
   892
      unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786
   893
      by (auto simp: positive_integral_density integrable_def borel_measurable_ereal_iff fin positive_integral_max_0
hoelzl@49786
   894
              cong: positive_integral_cong)
hoelzl@49786
   895
    have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) =
hoelzl@49786
   896
      integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786
   897
      using Px
hoelzl@49786
   898
      by (intro integrable_cong_AE)
hoelzl@49786
   899
         (auto simp: borel_measurable_ereal_iff log_divide_eq
hoelzl@49786
   900
                  intro!: measurable_If)
hoelzl@49786
   901
    then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))"
hoelzl@49786
   902
      unfolding distributed_distr_eq_density[OF X]
hoelzl@49786
   903
      using Px int
hoelzl@49786
   904
      by (subst integral_density) (auto simp: borel_measurable_ereal_iff)
hoelzl@49786
   905
  qed (auto simp: minus_log_convex[OF b_gt_1])
hoelzl@49786
   906
  also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)"
hoelzl@49786
   907
    unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786
   908
    by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq)
hoelzl@49786
   909
  also have "\<dots> = - entropy b MX X"
hoelzl@49786
   910
    unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786
   911
    by (subst entropy_distr[OF X]) (auto simp: borel_measurable_ereal_iff integral_density)
hoelzl@49786
   912
  finally show ?thesis
hoelzl@49786
   913
    by simp
hoelzl@49786
   914
qed
hoelzl@49786
   915
hoelzl@49786
   916
lemma (in information_space) entropy_le_space:
hoelzl@49786
   917
  fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
hoelzl@49786
   918
  assumes X: "distributed M MX X Px"
hoelzl@49786
   919
  and fin: "finite_measure MX"
hoelzl@49786
   920
  and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786
   921
  shows "entropy b MX X \<le> log b (measure MX (space MX))"
hoelzl@49786
   922
proof -
hoelzl@49786
   923
  note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786
   924
  interpret finite_measure MX by fact
hoelzl@49786
   925
  have "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
hoelzl@49786
   926
    using int X by (intro entropy_le) auto
hoelzl@49786
   927
  also have "\<dots> \<le> log b (measure MX (space MX))"
hoelzl@49786
   928
    using Px distributed_imp_emeasure_nonzero[OF X]
hoelzl@49786
   929
    by (intro log_le)
hoelzl@49786
   930
       (auto intro!: borel_measurable_ereal_iff finite_measure_mono b_gt_1
hoelzl@49786
   931
                     less_le[THEN iffD2] measure_nonneg simp: emeasure_eq_measure)
hoelzl@49786
   932
  finally show ?thesis .
hoelzl@49786
   933
qed
hoelzl@49786
   934
hoelzl@47694
   935
lemma (in information_space) entropy_uniform:
hoelzl@49785
   936
  assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f")
hoelzl@47694
   937
  shows "entropy b MX X = log b (measure MX A)"
hoelzl@49785
   938
proof (subst entropy_distr[OF X])
hoelzl@49785
   939
  have [simp]: "emeasure MX A \<noteq> \<infinity>"
hoelzl@49785
   940
    using uniform_distributed_params[OF X] by (auto simp add: measure_def)
hoelzl@49785
   941
  have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
hoelzl@49785
   942
    (\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)"
hoelzl@49785
   943
    using measure_nonneg[of MX A] uniform_distributed_params[OF X]
hoelzl@49785
   944
    by (auto intro!: integral_cong split: split_indicator simp: log_divide_eq)
hoelzl@49785
   945
  show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
hoelzl@49785
   946
    log b (measure MX A)"
hoelzl@49785
   947
    unfolding eq using uniform_distributed_params[OF X]
hoelzl@49785
   948
    by (subst lebesgue_integral_cmult) (auto simp: measure_def)
hoelzl@49785
   949
qed
hoelzl@36080
   950
hoelzl@47694
   951
lemma (in information_space) entropy_simple_distributed:
hoelzl@49786
   952
  "simple_distributed M X f \<Longrightarrow> \<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))"
hoelzl@49786
   953
  by (subst entropy_distr[OF simple_distributed])
hoelzl@49786
   954
     (auto simp add: lebesgue_integral_count_space_finite)
hoelzl@39097
   955
hoelzl@40859
   956
lemma (in information_space) entropy_le_card_not_0:
hoelzl@47694
   957
  assumes X: "simple_distributed M X f"
hoelzl@47694
   958
  shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))"
hoelzl@39097
   959
proof -
hoelzl@49787
   960
  let ?X = "count_space (X`space M)"
hoelzl@49787
   961
  have "\<H>(X) \<le> log b (measure ?X {x \<in> space ?X. f x \<noteq> 0})"
hoelzl@49787
   962
    by (rule entropy_le[OF simple_distributed[OF X]])
hoelzl@49787
   963
       (simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space)
hoelzl@49787
   964
  also have "measure ?X {x \<in> space ?X. f x \<noteq> 0} = card (X ` space M \<inter> {x. f x \<noteq> 0})"
hoelzl@49787
   965
    by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def Int_def)
hoelzl@49787
   966
  finally show ?thesis .
hoelzl@39097
   967
qed
hoelzl@39097
   968
hoelzl@40859
   969
lemma (in information_space) entropy_le_card:
hoelzl@49787
   970
  assumes X: "simple_distributed M X f"
hoelzl@40859
   971
  shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
hoelzl@49787
   972
proof -
hoelzl@49787
   973
  let ?X = "count_space (X`space M)"
hoelzl@49787
   974
  have "\<H>(X) \<le> log b (measure ?X (space ?X))"
hoelzl@49787
   975
    by (rule entropy_le_space[OF simple_distributed[OF X]])
hoelzl@49787
   976
       (simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space finite_measure_count_space)
hoelzl@49787
   977
  also have "measure ?X (space ?X) = card (X ` space M)"
hoelzl@49787
   978
    by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def)
hoelzl@39097
   979
  finally show ?thesis .
hoelzl@39097
   980
qed
hoelzl@39097
   981
hoelzl@39097
   982
subsection {* Conditional Mutual Information *}
hoelzl@39097
   983
hoelzl@36080
   984
definition (in prob_space)
hoelzl@41689
   985
  "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
hoelzl@41689
   986
    mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
hoelzl@41689
   987
    mutual_information b MX MZ X Z"
hoelzl@36080
   988
hoelzl@40859
   989
abbreviation (in information_space)
hoelzl@40859
   990
  conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
hoelzl@36624
   991
  "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
hoelzl@47694
   992
    (count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z"
hoelzl@36080
   993
hoelzl@49787
   994
lemma (in information_space)
hoelzl@47694
   995
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
hoelzl@50003
   996
  assumes Px[measurable]: "distributed M S X Px"
hoelzl@50003
   997
  assumes Pz[measurable]: "distributed M P Z Pz"
hoelzl@50003
   998
  assumes Pyz[measurable]: "distributed M (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
hoelzl@50003
   999
  assumes Pxz[measurable]: "distributed M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz"
hoelzl@50003
  1000
  assumes Pxyz[measurable]: "distributed M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
hoelzl@47694
  1001
  assumes I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
hoelzl@47694
  1002
  assumes I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
hoelzl@49787
  1003
  shows conditional_mutual_information_generic_eq: "conditional_mutual_information b S T P X Y Z
hoelzl@49787
  1004
    = (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" (is "?eq")
hoelzl@49787
  1005
    and conditional_mutual_information_generic_nonneg: "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
hoelzl@40859
  1006
proof -
hoelzl@47694
  1007
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
  1008
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
  1009
  interpret P: sigma_finite_measure P by fact
hoelzl@47694
  1010
  interpret TP: pair_sigma_finite T P ..
hoelzl@47694
  1011
  interpret SP: pair_sigma_finite S P ..
hoelzl@49787
  1012
  interpret ST: pair_sigma_finite S T ..
hoelzl@47694
  1013
  interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T ..
hoelzl@47694
  1014
  interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" ..
hoelzl@49787
  1015
  interpret TPS: pair_sigma_finite "T \<Otimes>\<^isub>M P" S ..
hoelzl@47694
  1016
  have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" ..
hoelzl@47694
  1017
  have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" ..
hoelzl@47694
  1018
  have YZ: "random_variable (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x))"
hoelzl@47694
  1019
    using Pyz by (simp add: distributed_measurable)
hoelzl@47694
  1020
  
hoelzl@47694
  1021
  from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) =
hoelzl@47694
  1022
    distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))"
hoelzl@50003
  1023
    by (simp add: comp_def distr_distr)
hoelzl@40859
  1024
hoelzl@47694
  1025
  have "mutual_information b S P X Z =
hoelzl@47694
  1026
    (\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))"
hoelzl@47694
  1027
    by (rule mutual_information_distr[OF S P Px Pz Pxz])
hoelzl@47694
  1028
  also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
hoelzl@47694
  1029
    using b_gt_1 Pxz Px Pz
hoelzl@50003
  1030
    by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) (auto simp: split_beta')
hoelzl@47694
  1031
  finally have mi_eq:
hoelzl@47694
  1032
    "mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" .
hoelzl@47694
  1033
  
hoelzl@49787
  1034
  have ae1: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@47694
  1035
    by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
hoelzl@49787
  1036
  moreover have ae2: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1037
    by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
hoelzl@49787
  1038
  moreover have ae3: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1039
    by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
hoelzl@49787
  1040
  moreover have ae4: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1041
    by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
hoelzl@49787
  1042
  moreover have ae5: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)"
hoelzl@50003
  1043
    using Px by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
hoelzl@49787
  1044
  moreover have ae6: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)"
hoelzl@50003
  1045
    using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
hoelzl@49787
  1046
  moreover have ae7: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))"
hoelzl@50003
  1047
    using Pz Pz[THEN distributed_real_measurable]
hoelzl@50003
  1048
    by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
hoelzl@49787
  1049
  moreover have ae8: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
hoelzl@47694
  1050
    using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
hoelzl@50003
  1051
    by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure)
hoelzl@47694
  1052
  moreover note Pxyz[THEN distributed_real_AE]
hoelzl@49787
  1053
  ultimately have ae: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P.
hoelzl@47694
  1054
    Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
hoelzl@47694
  1055
    Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
hoelzl@47694
  1056
    Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
hoelzl@47694
  1057
  proof eventually_elim
hoelzl@47694
  1058
    case (goal1 x)
hoelzl@47694
  1059
    show ?case
hoelzl@40859
  1060
    proof cases
hoelzl@47694
  1061
      assume "Pxyz x \<noteq> 0"
hoelzl@47694
  1062
      with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x"
hoelzl@47694
  1063
        by auto
hoelzl@47694
  1064
      then show ?thesis
hoelzl@47694
  1065
        using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@40859
  1066
    qed simp
hoelzl@40859
  1067
  qed
hoelzl@49787
  1068
  with I1 I2 show ?eq
hoelzl@40859
  1069
    unfolding conditional_mutual_information_def
hoelzl@47694
  1070
    apply (subst mi_eq)
hoelzl@47694
  1071
    apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
hoelzl@47694
  1072
    apply (subst integral_diff(2)[symmetric])
hoelzl@47694
  1073
    apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
hoelzl@47694
  1074
    done
hoelzl@49787
  1075
hoelzl@49787
  1076
  let ?P = "density (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) Pxyz"
hoelzl@49787
  1077
  interpret P: prob_space ?P
hoelzl@49787
  1078
    unfolding distributed_distr_eq_density[OF Pxyz, symmetric]
hoelzl@50003
  1079
    by (rule prob_space_distr) simp
hoelzl@49787
  1080
hoelzl@49787
  1081
  let ?Q = "density (T \<Otimes>\<^isub>M P) Pyz"
hoelzl@49787
  1082
  interpret Q: prob_space ?Q
hoelzl@49787
  1083
    unfolding distributed_distr_eq_density[OF Pyz, symmetric]
hoelzl@50003
  1084
    by (rule prob_space_distr) simp
hoelzl@49787
  1085
hoelzl@49787
  1086
  let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"
hoelzl@49787
  1087
hoelzl@49787
  1088
  from subdensity_real[of snd, OF _ Pyz Pz]
hoelzl@49787
  1089
  have aeX1: "AE x in T \<Otimes>\<^isub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
hoelzl@49787
  1090
  have aeX2: "AE x in T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd x)"
hoelzl@50003
  1091
    using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
hoelzl@49787
  1092
hoelzl@49787
  1093
  have aeX3: "AE y in T \<Otimes>\<^isub>M P. (\<integral>\<^isup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
hoelzl@49788
  1094
    using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
hoelzl@50003
  1095
    by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49787
  1096
hoelzl@49787
  1097
  have "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^isup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
hoelzl@49787
  1098
    apply (subst positive_integral_density)
hoelzl@50003
  1099
    apply simp
hoelzl@49787
  1100
    apply (rule distributed_AE[OF Pxyz])
hoelzl@50003
  1101
    apply auto []
hoelzl@49787
  1102
    apply (rule positive_integral_mono_AE)
hoelzl@49787
  1103
    using ae5 ae6 ae7 ae8
hoelzl@49787
  1104
    apply eventually_elim
hoelzl@49787
  1105
    apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
hoelzl@49787
  1106
    done
hoelzl@49787
  1107
  also have "\<dots> = (\<integral>\<^isup>+(y, z). \<integral>\<^isup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^isub>M P)"
hoelzl@50003
  1108
    by (subst STP.positive_integral_snd_measurable[symmetric]) (auto simp add: split_beta')
hoelzl@49787
  1109
  also have "\<dots> = (\<integral>\<^isup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^isub>M P)"
hoelzl@49787
  1110
    apply (rule positive_integral_cong_AE)
hoelzl@49787
  1111
    using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
hoelzl@49787
  1112
    apply eventually_elim
hoelzl@49787
  1113
  proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
hoelzl@49787
  1114
    fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P"
hoelzl@49787
  1115
      "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" 
hoelzl@49787
  1116
    then show "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
hoelzl@50003
  1117
      by (subst positive_integral_multc)
hoelzl@50003
  1118
         (auto intro!: divide_nonneg_nonneg split: prod.split)
hoelzl@49787
  1119
  qed
hoelzl@49787
  1120
  also have "\<dots> = 1"
hoelzl@49787
  1121
    using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
hoelzl@50003
  1122
    by (subst positive_integral_density[symmetric]) auto
hoelzl@49787
  1123
  finally have le1: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> 1" .
hoelzl@49787
  1124
  also have "\<dots> < \<infinity>" by simp
hoelzl@49787
  1125
  finally have fin: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp
hoelzl@49787
  1126
hoelzl@49787
  1127
  have pos: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> 0"
hoelzl@49787
  1128
    apply (subst positive_integral_density)
hoelzl@50003
  1129
    apply simp
hoelzl@49787
  1130
    apply (rule distributed_AE[OF Pxyz])
hoelzl@50003
  1131
    apply auto []
hoelzl@49787
  1132
    apply (simp add: split_beta')
hoelzl@49787
  1133
  proof
hoelzl@49787
  1134
    let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))"
hoelzl@49787
  1135
    assume "(\<integral>\<^isup>+ x. ?g x \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)) = 0"
hoelzl@49787
  1136
    then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. ?g x \<le> 0"
hoelzl@50003
  1137
      by (intro positive_integral_0_iff_AE[THEN iffD1]) auto
hoelzl@49787
  1138
    then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxyz x = 0"
hoelzl@49787
  1139
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787
  1140
      by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
hoelzl@49787
  1141
    then have "(\<integral>\<^isup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) = 0"
hoelzl@49787
  1142
      by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
hoelzl@49787
  1143
    with P.emeasure_space_1 show False
hoelzl@50003
  1144
      by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
hoelzl@49787
  1145
  qed
hoelzl@49787
  1146
hoelzl@49787
  1147
  have neg: "(\<integral>\<^isup>+ x. - ?f x \<partial>?P) = 0"
hoelzl@49787
  1148
    apply (rule positive_integral_0_iff_AE[THEN iffD2])
hoelzl@50003
  1149
    apply simp
hoelzl@49787
  1150
    apply (subst AE_density)
hoelzl@50003
  1151
    apply simp
hoelzl@49787
  1152
    using ae5 ae6 ae7 ae8
hoelzl@49787
  1153
    apply eventually_elim
hoelzl@49787
  1154
    apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
hoelzl@49787
  1155
    done
hoelzl@49787
  1156
hoelzl@49787
  1157
  have I3: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
hoelzl@49787
  1158
    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
hoelzl@49787
  1159
    using ae
hoelzl@50003
  1160
    apply (auto simp: split_beta')
hoelzl@49787
  1161
    done
hoelzl@49787
  1162
hoelzl@49787
  1163
  have "- log b 1 \<le> - log b (integral\<^isup>L ?P ?f)"
hoelzl@49787
  1164
  proof (intro le_imp_neg_le log_le[OF b_gt_1])
hoelzl@49787
  1165
    show "0 < integral\<^isup>L ?P ?f"
hoelzl@49787
  1166
      using neg pos fin positive_integral_positive[of ?P ?f]
hoelzl@49787
  1167
      by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
hoelzl@49787
  1168
    show "integral\<^isup>L ?P ?f \<le> 1"
hoelzl@49787
  1169
      using neg le1 fin positive_integral_positive[of ?P ?f]
hoelzl@49787
  1170
      by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
hoelzl@49787
  1171
  qed
hoelzl@49787
  1172
  also have "- log b (integral\<^isup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
hoelzl@49787
  1173
  proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
hoelzl@49787
  1174
    show "AE x in ?P. ?f x \<in> {0<..}"
hoelzl@49787
  1175
      unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
hoelzl@49787
  1176
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787
  1177
      by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
hoelzl@49787
  1178
    show "integrable ?P ?f"
hoelzl@49787
  1179
      unfolding integrable_def 
hoelzl@50003
  1180
      using fin neg by (auto simp: split_beta')
hoelzl@49787
  1181
    show "integrable ?P (\<lambda>x. - log b (?f x))"
hoelzl@49787
  1182
      apply (subst integral_density)
hoelzl@50003
  1183
      apply simp
hoelzl@50003
  1184
      apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003
  1185
      apply simp
hoelzl@49787
  1186
      apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
hoelzl@50003
  1187
      apply simp
hoelzl@50003
  1188
      apply simp
hoelzl@49787
  1189
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787
  1190
      apply eventually_elim
hoelzl@49787
  1191
      apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49787
  1192
      done
hoelzl@49787
  1193
  qed (auto simp: b_gt_1 minus_log_convex)
hoelzl@49787
  1194
  also have "\<dots> = conditional_mutual_information b S T P X Y Z"
hoelzl@49787
  1195
    unfolding `?eq`
hoelzl@49787
  1196
    apply (subst integral_density)
hoelzl@50003
  1197
    apply simp
hoelzl@50003
  1198
    apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003
  1199
    apply simp
hoelzl@49787
  1200
    apply (intro integral_cong_AE)
hoelzl@49787
  1201
    using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787
  1202
    apply eventually_elim
hoelzl@49787
  1203
    apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49787
  1204
    done
hoelzl@49787
  1205
  finally show ?nonneg
hoelzl@49787
  1206
    by simp
hoelzl@40859
  1207
qed
hoelzl@40859
  1208
hoelzl@49803
  1209
lemma (in information_space)
hoelzl@49803
  1210
  fixes Px :: "_ \<Rightarrow> real"
hoelzl@49803
  1211
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
hoelzl@49803
  1212
  assumes Fx: "finite_entropy S X Px"
hoelzl@49803
  1213
  assumes Fz: "finite_entropy P Z Pz"
hoelzl@49803
  1214
  assumes Fyz: "finite_entropy (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
hoelzl@49803
  1215
  assumes Fxz: "finite_entropy (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz"
hoelzl@49803
  1216
  assumes Fxyz: "finite_entropy (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
hoelzl@49803
  1217
  shows conditional_mutual_information_generic_eq': "conditional_mutual_information b S T P X Y Z
hoelzl@49803
  1218
    = (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" (is "?eq")
hoelzl@49803
  1219
    and conditional_mutual_information_generic_nonneg': "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
hoelzl@49803
  1220
proof -
hoelzl@50003
  1221
  note Px = Fx[THEN finite_entropy_distributed, measurable]
hoelzl@50003
  1222
  note Pz = Fz[THEN finite_entropy_distributed, measurable]
hoelzl@50003
  1223
  note Pyz = Fyz[THEN finite_entropy_distributed, measurable]
hoelzl@50003
  1224
  note Pxz = Fxz[THEN finite_entropy_distributed, measurable]
hoelzl@50003
  1225
  note Pxyz = Fxyz[THEN finite_entropy_distributed, measurable]
hoelzl@49803
  1226
hoelzl@49803
  1227
  interpret S: sigma_finite_measure S by fact
hoelzl@49803
  1228
  interpret T: sigma_finite_measure T by fact
hoelzl@49803
  1229
  interpret P: sigma_finite_measure P by fact
hoelzl@49803
  1230
  interpret TP: pair_sigma_finite T P ..
hoelzl@49803
  1231
  interpret SP: pair_sigma_finite S P ..
hoelzl@49803
  1232
  interpret ST: pair_sigma_finite S T ..
hoelzl@49803
  1233
  interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T ..
hoelzl@49803
  1234
  interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" ..
hoelzl@49803
  1235
  interpret TPS: pair_sigma_finite "T \<Otimes>\<^isub>M P" S ..
hoelzl@49803
  1236
  have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" ..
hoelzl@49803
  1237
  have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" ..
hoelzl@49803
  1238
hoelzl@49803
  1239
  from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) =
hoelzl@49803
  1240
    distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))"
hoelzl@50003
  1241
    by (simp add: distr_distr comp_def)
hoelzl@49803
  1242
hoelzl@49803
  1243
  have "mutual_information b S P X Z =
hoelzl@49803
  1244
    (\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))"
hoelzl@49803
  1245
    by (rule mutual_information_distr[OF S P Px Pz Pxz])
hoelzl@49803
  1246
  also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
hoelzl@49803
  1247
    using b_gt_1 Pxz Px Pz
hoelzl@49803
  1248
    by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"])
hoelzl@50003
  1249
       (auto simp: split_beta')
hoelzl@49803
  1250
  finally have mi_eq:
hoelzl@49803
  1251
    "mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" .
hoelzl@49803
  1252
  
hoelzl@49803
  1253
  have ae1: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@49803
  1254
    by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
hoelzl@49803
  1255
  moreover have ae2: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1256
    by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
hoelzl@49803
  1257
  moreover have ae3: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1258
    by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
hoelzl@49803
  1259
  moreover have ae4: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003
  1260
    by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
hoelzl@49803
  1261
  moreover have ae5: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)"
hoelzl@50003
  1262
    using Px by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49803
  1263
  moreover have ae6: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)"
hoelzl@50003
  1264
    using Pyz by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49803
  1265
  moreover have ae7: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))"
hoelzl@50003
  1266
    using Pz Pz[THEN distributed_real_measurable] by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
hoelzl@49803
  1267
  moreover have ae8: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
hoelzl@49803
  1268
    using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
hoelzl@49803
  1269
    by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def)
hoelzl@49803
  1270
  moreover note ae9 = Pxyz[THEN distributed_real_AE]
hoelzl@49803
  1271
  ultimately have ae: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P.
hoelzl@49803
  1272
    Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
hoelzl@49803
  1273
    Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
hoelzl@49803
  1274
    Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
hoelzl@49803
  1275
  proof eventually_elim
hoelzl@49803
  1276
    case (goal1 x)
hoelzl@49803
  1277
    show ?case
hoelzl@49803
  1278
    proof cases
hoelzl@49803
  1279
      assume "Pxyz x \<noteq> 0"
hoelzl@49803
  1280
      with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x"
hoelzl@49803
  1281
        by auto
hoelzl@49803
  1282
      then show ?thesis
hoelzl@49803
  1283
        using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@49803
  1284
    qed simp
hoelzl@49803
  1285
  qed
hoelzl@49803
  1286
hoelzl@49803
  1287
  have "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)
hoelzl@49803
  1288
    (\<lambda>x. Pxyz x * log b (Pxyz x) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pyz (snd x)))"
hoelzl@49803
  1289
    using finite_entropy_integrable[OF Fxyz]
hoelzl@49803
  1290
    using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
hoelzl@49803
  1291
    using finite_entropy_integrable_transform[OF Fyz Pxyz, of snd]
hoelzl@49803
  1292
    by simp
hoelzl@49803
  1293
  moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z)))) \<in> borel_measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)"
hoelzl@50003
  1294
    using Pxyz Px Pyz by simp
hoelzl@49803
  1295
  ultimately have I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
hoelzl@49803
  1296
    apply (rule integrable_cong_AE_imp)
hoelzl@49803
  1297
    using ae1 ae4 ae5 ae6 ae9
hoelzl@49803
  1298
    by eventually_elim
hoelzl@49803
  1299
       (auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)
hoelzl@49803
  1300
hoelzl@49803
  1301
  have "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)
hoelzl@49803
  1302
    (\<lambda>x. Pxyz x * log b (Pxz (fst x, snd (snd x))) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pz (snd (snd x))))"
hoelzl@49803
  1303
    using finite_entropy_integrable_transform[OF Fxz Pxyz, of "\<lambda>x. (fst x, snd (snd x))"]
hoelzl@49803
  1304
    using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
hoelzl@49803
  1305
    using finite_entropy_integrable_transform[OF Fz Pxyz, of "snd \<circ> snd"]
hoelzl@50003
  1306
    by simp
hoelzl@49803
  1307
  moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z))) \<in> borel_measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)"
hoelzl@49803
  1308
    using Pxyz Px Pz
hoelzl@50003
  1309
    by auto
hoelzl@49803
  1310
  ultimately have I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
hoelzl@49803
  1311
    apply (rule integrable_cong_AE_imp)
hoelzl@49803
  1312
    using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 ae9
hoelzl@49803
  1313
    by eventually_elim
hoelzl@49803
  1314
       (auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)
hoelzl@49803
  1315
hoelzl@49803
  1316
  from ae I1 I2 show ?eq
hoelzl@49803
  1317
    unfolding conditional_mutual_information_def
hoelzl@49803
  1318
    apply (subst mi_eq)
hoelzl@49803
  1319
    apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
hoelzl@49803
  1320
    apply (subst integral_diff(2)[symmetric])
hoelzl@49803
  1321
    apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
hoelzl@49803
  1322
    done
hoelzl@49803
  1323
hoelzl@49803
  1324
  let ?P = "density (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) Pxyz"
hoelzl@49803
  1325
  interpret P: prob_space ?P
hoelzl@50003
  1326
    unfolding distributed_distr_eq_density[OF Pxyz, symmetric] by (rule prob_space_distr) simp
hoelzl@49803
  1327
hoelzl@49803
  1328
  let ?Q = "density (T \<Otimes>\<^isub>M P) Pyz"
hoelzl@49803
  1329
  interpret Q: prob_space ?Q
hoelzl@50003
  1330
    unfolding distributed_distr_eq_density[OF Pyz, symmetric] by (rule prob_space_distr) simp
hoelzl@49803
  1331
hoelzl@49803
  1332
  let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"
hoelzl@49803
  1333
hoelzl@49803
  1334
  from subdensity_real[of snd, OF _ Pyz Pz]
hoelzl@49803
  1335
  have aeX1: "AE x in T \<Otimes>\<^isub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
hoelzl@49803
  1336
  have aeX2: "AE x in T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd x)"
hoelzl@50003
  1337
    using Pz by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49803
  1338
hoelzl@49803
  1339
  have aeX3: "AE y in T \<Otimes>\<^isub>M P. (\<integral>\<^isup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
hoelzl@49803
  1340
    using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
hoelzl@50003
  1341
    by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49803
  1342
  have "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^isup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
hoelzl@49803
  1343
    apply (subst positive_integral_density)
hoelzl@49803
  1344
    apply (rule distributed_borel_measurable[OF Pxyz])
hoelzl@49803
  1345
    apply (rule distributed_AE[OF Pxyz])
hoelzl@50003
  1346
    apply simp
hoelzl@49803
  1347
    apply (rule positive_integral_mono_AE)
hoelzl@49803
  1348
    using ae5 ae6 ae7 ae8
hoelzl@49803
  1349
    apply eventually_elim
hoelzl@49803
  1350
    apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
hoelzl@49803
  1351
    done
hoelzl@49803
  1352
  also have "\<dots> = (\<integral>\<^isup>+(y, z). \<integral>\<^isup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^isub>M P)"
hoelzl@49803
  1353
    by (subst STP.positive_integral_snd_measurable[symmetric])
hoelzl@50003
  1354
       (auto simp add: split_beta')
hoelzl@49803
  1355
  also have "\<dots> = (\<integral>\<^isup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^isub>M P)"
hoelzl@49803
  1356
    apply (rule positive_integral_cong_AE)
hoelzl@49803
  1357
    using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
hoelzl@49803
  1358
    apply eventually_elim
hoelzl@49803
  1359
  proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
hoelzl@49803
  1360
    fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P"
hoelzl@49803
  1361
      "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" 
hoelzl@49803
  1362
    then show "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
hoelzl@50003
  1363
      by (subst positive_integral_multc) (auto intro!: divide_nonneg_nonneg)
hoelzl@49803
  1364
  qed
hoelzl@49803
  1365
  also have "\<dots> = 1"
hoelzl@49803
  1366
    using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
hoelzl@50003
  1367
    by (subst positive_integral_density[symmetric]) auto
hoelzl@49803
  1368
  finally have le1: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> 1" .
hoelzl@49803
  1369
  also have "\<dots> < \<infinity>" by simp
hoelzl@49803
  1370
  finally have fin: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp
hoelzl@49803
  1371
hoelzl@49803
  1372
  have pos: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> 0"
hoelzl@49803
  1373
    apply (subst positive_integral_density)
hoelzl@50003
  1374
    apply simp
hoelzl@49803
  1375
    apply (rule distributed_AE[OF Pxyz])
hoelzl@50003
  1376
    apply simp
hoelzl@49803
  1377
    apply (simp add: split_beta')
hoelzl@49803
  1378
  proof
hoelzl@49803
  1379
    let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))"
hoelzl@49803
  1380
    assume "(\<integral>\<^isup>+ x. ?g x \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)) = 0"
hoelzl@49803
  1381
    then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. ?g x \<le> 0"
hoelzl@50003
  1382
      by (intro positive_integral_0_iff_AE[THEN iffD1]) (auto intro!: borel_measurable_ereal measurable_If)
hoelzl@49803
  1383
    then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxyz x = 0"
hoelzl@49803
  1384
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803
  1385
      by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
hoelzl@49803
  1386
    then have "(\<integral>\<^isup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) = 0"
hoelzl@49803
  1387
      by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
hoelzl@49803
  1388
    with P.emeasure_space_1 show False
hoelzl@50003
  1389
      by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
hoelzl@49803
  1390
  qed
hoelzl@49803
  1391
hoelzl@49803
  1392
  have neg: "(\<integral>\<^isup>+ x. - ?f x \<partial>?P) = 0"
hoelzl@49803
  1393
    apply (rule positive_integral_0_iff_AE[THEN iffD2])
hoelzl@50003
  1394
    apply (auto simp: split_beta') []
hoelzl@49803
  1395
    apply (subst AE_density)
hoelzl@50003
  1396
    apply (auto simp: split_beta') []
hoelzl@49803
  1397
    using ae5 ae6 ae7 ae8
hoelzl@49803
  1398
    apply eventually_elim
hoelzl@49803
  1399
    apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
hoelzl@49803
  1400
    done
hoelzl@49803
  1401
hoelzl@49803
  1402
  have I3: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
hoelzl@49803
  1403
    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
hoelzl@49803
  1404
    using ae
hoelzl@50003
  1405
    apply (auto simp: split_beta')
hoelzl@49803
  1406
    done
hoelzl@49803
  1407
hoelzl@49803
  1408
  have "- log b 1 \<le> - log b (integral\<^isup>L ?P ?f)"
hoelzl@49803
  1409
  proof (intro le_imp_neg_le log_le[OF b_gt_1])
hoelzl@49803
  1410
    show "0 < integral\<^isup>L ?P ?f"
hoelzl@49803
  1411
      using neg pos fin positive_integral_positive[of ?P ?f]
hoelzl@49803
  1412
      by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
hoelzl@49803
  1413
    show "integral\<^isup>L ?P ?f \<le> 1"
hoelzl@49803
  1414
      using neg le1 fin positive_integral_positive[of ?P ?f]
hoelzl@49803
  1415
      by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
hoelzl@49803
  1416
  qed
hoelzl@49803
  1417
  also have "- log b (integral\<^isup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
hoelzl@49803
  1418
  proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
hoelzl@49803
  1419
    show "AE x in ?P. ?f x \<in> {0<..}"
hoelzl@49803
  1420
      unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
hoelzl@49803
  1421
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803
  1422
      by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
hoelzl@49803
  1423
    show "integrable ?P ?f"
hoelzl@49803
  1424
      unfolding integrable_def 
hoelzl@50003
  1425
      using fin neg by (auto simp: split_beta')
hoelzl@49803
  1426
    show "integrable ?P (\<lambda>x. - log b (?f x))"
hoelzl@49803
  1427
      apply (subst integral_density)
hoelzl@50003
  1428
      apply simp
hoelzl@50003
  1429
      apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003
  1430
      apply simp
hoelzl@49803
  1431
      apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
hoelzl@50003
  1432
      apply simp
hoelzl@50003
  1433
      apply simp
hoelzl@49803
  1434
      using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803
  1435
      apply eventually_elim
hoelzl@49803
  1436
      apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49803
  1437
      done
hoelzl@49803
  1438
  qed (auto simp: b_gt_1 minus_log_convex)
hoelzl@49803
  1439
  also have "\<dots> = conditional_mutual_information b S T P X Y Z"
hoelzl@49803
  1440
    unfolding `?eq`
hoelzl@49803
  1441
    apply (subst integral_density)
hoelzl@50003
  1442
    apply simp
hoelzl@50003
  1443
    apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003
  1444
    apply simp
hoelzl@49803
  1445
    apply (intro integral_cong_AE)
hoelzl@49803
  1446
    using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803
  1447
    apply eventually_elim
hoelzl@49803
  1448
    apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49803
  1449
    done
hoelzl@49803
  1450
  finally show ?nonneg
hoelzl@49803
  1451
    by simp
hoelzl@49803
  1452
qed
hoelzl@49803
  1453
hoelzl@40859
  1454
lemma (in information_space) conditional_mutual_information_eq:
hoelzl@47694
  1455
  assumes Pz: "simple_distributed M Z Pz"
hoelzl@47694
  1456
  assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
hoelzl@47694
  1457
  assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
hoelzl@47694
  1458
  assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
hoelzl@47694
  1459
  shows "\<I>(X ; Y | Z) =
hoelzl@47694
  1460
   (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x))`space M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
hoelzl@47694
  1461
proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _
hoelzl@47694
  1462
    simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz]
hoelzl@47694
  1463
    simple_distributed_joint2[OF Pxyz]])
hoelzl@47694
  1464
  note simple_distributed_joint2_finite[OF Pxyz, simp]
hoelzl@47694
  1465
  show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694
  1466
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
  1467
  show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694
  1468
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
  1469
  show "sigma_finite_measure (count_space (Z ` space M))"
hoelzl@47694
  1470
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
  1471
  have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) =
hoelzl@47694
  1472
      count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@47694
  1473
    (is "?P = ?C")
hoelzl@47694
  1474
    by (simp add: pair_measure_count_space)
hoelzl@40859
  1475
hoelzl@47694
  1476
  let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)"
hoelzl@47694
  1477
  have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^isub>M count_space (Z ` space M))"
hoelzl@47694
  1478
    using simple_distributed_joint[OF Pxz] by (rule distributed_measurable)
hoelzl@47694
  1479
  from measurable_comp[OF this measurable_fst]
hoelzl@47694
  1480
  have "random_variable (count_space (X ` space M)) X"
hoelzl@47694
  1481
    by (simp add: comp_def)
hoelzl@47694
  1482
  then have "simple_function M X"    
hoelzl@50002
  1483
    unfolding simple_function_def by (auto simp: measurable_count_space_eq2)
hoelzl@47694
  1484
  then have "simple_distributed M X ?Px"
hoelzl@47694
  1485
    by (rule simple_distributedI) auto
hoelzl@47694
  1486
  then show "distributed M (count_space (X ` space M)) X ?Px"
hoelzl@47694
  1487
    by (rule simple_distributed)
hoelzl@47694
  1488
hoelzl@47694
  1489
  let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)"
hoelzl@47694
  1490
  let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)"
hoelzl@47694
  1491
  let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)"
hoelzl@47694
  1492
  show
hoelzl@47694
  1493
      "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))"
hoelzl@47694
  1494
      "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))"
hoelzl@47694
  1495
    by (auto intro!: integrable_count_space simp: pair_measure_count_space)
hoelzl@47694
  1496
  let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))"
hoelzl@47694
  1497
  let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))"
hoelzl@47694
  1498
  have "(\<lambda>(x, y, z). ?i x y z) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then ?j (fst x) (fst (snd x)) (snd (snd x)) else 0)"
hoelzl@47694
  1499
    by (auto intro!: ext)
hoelzl@47694
  1500
  then show "(\<integral> (x, y, z). ?i x y z \<partial>?P) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. ?j x y z)"
hoelzl@47694
  1501
    by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta')
hoelzl@36624
  1502
qed
hoelzl@36624
  1503
hoelzl@47694
  1504
lemma (in information_space) conditional_mutual_information_nonneg:
hoelzl@47694
  1505
  assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z"
hoelzl@47694
  1506
  shows "0 \<le> \<I>(X ; Y | Z)"
hoelzl@47694
  1507
proof -
hoelzl@49787
  1508
  have [simp]: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) =
hoelzl@49787
  1509
      count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@49787
  1510
    by (simp add: pair_measure_count_space X Y Z simple_functionD)
hoelzl@49787
  1511
  note sf = sigma_finite_measure_count_space_finite[OF simple_functionD(1)]
hoelzl@49787
  1512
  note sd = simple_distributedI[OF _ refl]
hoelzl@49787
  1513
  note sp = simple_function_Pair
hoelzl@49787
  1514
  show ?thesis
hoelzl@49787
  1515
   apply (rule conditional_mutual_information_generic_nonneg[OF sf[OF X] sf[OF Y] sf[OF Z]])
hoelzl@49787
  1516
   apply (rule simple_distributed[OF sd[OF X]])
hoelzl@49787
  1517
   apply (rule simple_distributed[OF sd[OF Z]])
hoelzl@49787
  1518
   apply (rule simple_distributed_joint[OF sd[OF sp[OF Y Z]]])
hoelzl@49787
  1519
   apply (rule simple_distributed_joint[OF sd[OF sp[OF X Z]]])
hoelzl@49787
  1520
   apply (rule simple_distributed_joint2[OF sd[OF sp[OF X sp[OF Y Z]]]])
hoelzl@49787
  1521
   apply (auto intro!: integrable_count_space simp: X Y Z simple_functionD)
hoelzl@49787
  1522
   done
hoelzl@36080
  1523
qed
hoelzl@36080
  1524
hoelzl@39097
  1525
subsection {* Conditional Entropy *}
hoelzl@39097
  1526
hoelzl@36080
  1527
definition (in prob_space)
hoelzl@49791
  1528
  "conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real (RN_deriv (S \<Otimes>\<^isub>M T) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / 
hoelzl@49791
  1529
    real (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))"
hoelzl@36080
  1530
hoelzl@40859
  1531
abbreviation (in information_space)
hoelzl@40859
  1532
  conditional_entropy_Pow ("\<H>'(_ | _')") where
hoelzl@47694
  1533
  "\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y"
hoelzl@36080
  1534
hoelzl@49791
  1535
lemma (in information_space) conditional_entropy_generic_eq:
hoelzl@49791
  1536
  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@49791
  1537
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@50003
  1538
  assumes Py[measurable]: "distributed M T Y Py"
hoelzl@50003
  1539
  assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49791
  1540
  shows "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49791
  1541
proof -
hoelzl@49791
  1542
  interpret S: sigma_finite_measure S by fact
hoelzl@49791
  1543
  interpret T: sigma_finite_measure T by fact
hoelzl@49791
  1544
  interpret ST: pair_sigma_finite S T ..
hoelzl@49791
  1545
hoelzl@49791
  1546
  have "AE x in density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real (RN_deriv (S \<Otimes>\<^isub>M T) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) x)"
hoelzl@49791
  1547
    unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
hoelzl@49791
  1548
    unfolding distributed_distr_eq_density[OF Pxy]
hoelzl@49791
  1549
    using distributed_RN_deriv[OF Pxy]
hoelzl@49791
  1550
    by auto
hoelzl@49791
  1551
  moreover
hoelzl@49791
  1552
  have "AE x in density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real (RN_deriv T (distr M T Y) (snd x))"
hoelzl@49791
  1553
    unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
hoelzl@49791
  1554
    unfolding distributed_distr_eq_density[OF Py]
hoelzl@49791
  1555
    apply (rule ST.AE_pair_measure)
immler@50244
  1556
    apply (auto intro!: sets.sets_Collect borel_measurable_eq measurable_compose[OF _ distributed_real_measurable[OF Py]]
hoelzl@49791
  1557
                        distributed_real_measurable[OF Pxy] distributed_real_AE[OF Py]
hoelzl@49791
  1558
                        borel_measurable_real_of_ereal measurable_compose[OF _ borel_measurable_RN_deriv_density])
hoelzl@49791
  1559
    using distributed_RN_deriv[OF Py]
hoelzl@49791
  1560
    apply auto
hoelzl@49791
  1561
    done    
hoelzl@49791
  1562
  ultimately
hoelzl@49791
  1563
  have "conditional_entropy b S T X Y = - (\<integral>x. Pxy x * log b (Pxy x / Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49791
  1564
    unfolding conditional_entropy_def neg_equal_iff_equal
hoelzl@49791
  1565
    apply (subst integral_density(1)[symmetric])
hoelzl@49791
  1566
    apply (auto simp: distributed_real_measurable[OF Pxy] distributed_real_AE[OF Pxy]
hoelzl@49791
  1567
                      measurable_compose[OF _ distributed_real_measurable[OF Py]]
hoelzl@49791
  1568
                      distributed_distr_eq_density[OF Pxy]
hoelzl@49791
  1569
                intro!: integral_cong_AE)
hoelzl@49791
  1570
    done
hoelzl@49791
  1571
  then show ?thesis by (simp add: split_beta')
hoelzl@49791
  1572
qed
hoelzl@49791
  1573
hoelzl@49791
  1574
lemma (in information_space) conditional_entropy_eq_entropy:
hoelzl@47694
  1575
  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694
  1576
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@47694
  1577
  assumes Py: "distributed M T Y Py"
hoelzl@47694
  1578
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
  1579
  assumes I1: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
hoelzl@47694
  1580
  assumes I2: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
hoelzl@49791
  1581
  shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
hoelzl@40859
  1582
proof -
hoelzl@47694
  1583
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
  1584
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
  1585
  interpret ST: pair_sigma_finite S T ..
hoelzl@47694
  1586
hoelzl@47694
  1587
  have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)"
hoelzl@49786
  1588
    by (rule entropy_distr[OF Py])
hoelzl@47694
  1589
  also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@47694
  1590
    using b_gt_1 Py[THEN distributed_real_measurable]
hoelzl@47694
  1591
    by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong)
hoelzl@47694
  1592
  finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" .
hoelzl@49791
  1593
hoelzl@49790
  1594
  have ae2: "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
  1595
    by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
hoelzl@49788
  1596
  moreover have ae4: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
hoelzl@47694
  1597
    using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
hoelzl@49788
  1598
  moreover note ae5 = Pxy[THEN distributed_real_AE]
hoelzl@49791
  1599
  ultimately have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Pxy x \<and> 0 \<le> Py (snd x) \<and>
hoelzl@49790
  1600
    (Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Py (snd x)))"
hoelzl@47694
  1601
    by eventually_elim auto
hoelzl@49791
  1602
  then have ae: "AE x in S \<Otimes>\<^isub>M T.
hoelzl@47694
  1603
     Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))"
hoelzl@47694
  1604
    by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1)
hoelzl@49791
  1605
  have "conditional_entropy b S T X Y = 
hoelzl@49791
  1606
    - (\<integral>x. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49791
  1607
    unfolding conditional_entropy_generic_eq[OF S T Py Pxy] neg_equal_iff_equal
hoelzl@49791
  1608
    apply (intro integral_cong_AE)
hoelzl@49791
  1609
    using ae
hoelzl@49791
  1610
    apply eventually_elim
hoelzl@49791
  1611
    apply auto
hoelzl@47694
  1612
    done
hoelzl@49791
  1613
  also have "\<dots> = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) - - (\<integral>x.  Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49791
  1614
    by (simp add: integral_diff[OF I1 I2])
hoelzl@49791
  1615
  finally show ?thesis 
hoelzl@49791
  1616
    unfolding conditional_entropy_generic_eq[OF S T Py Pxy] entropy_distr[OF Pxy] e_eq
hoelzl@49791
  1617
    by (simp add: split_beta')
hoelzl@49791
  1618
qed
hoelzl@49791
  1619
hoelzl@49791
  1620
lemma (in information_space) conditional_entropy_eq_entropy_simple:
hoelzl@49791
  1621
  assumes X: "simple_function M X" and Y: "simple_function M Y"
hoelzl@49791
  1622
  shows "\<H>(X | Y) = entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x)) - \<H>(Y)"
hoelzl@49791
  1623
proof -
hoelzl@49791
  1624
  have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
hoelzl@49791
  1625
    (is "?P = ?C") using X Y by (simp add: simple_functionD pair_measure_count_space)
hoelzl@49791
  1626
  show ?thesis
hoelzl@49791
  1627
    by (rule conditional_entropy_eq_entropy sigma_finite_measure_count_space_finite
hoelzl@49791
  1628
                 simple_functionD  X Y simple_distributed simple_distributedI[OF _ refl]
hoelzl@49791
  1629
                 simple_distributed_joint simple_function_Pair integrable_count_space)+
hoelzl@49791
  1630
       (auto simp: `?P = ?C` intro!: integrable_count_space simple_functionD  X Y)
hoelzl@39097
  1631
qed
hoelzl@39097
  1632
hoelzl@40859
  1633
lemma (in information_space) conditional_entropy_eq:
hoelzl@49792
  1634
  assumes Y: "simple_distributed M Y Py"
hoelzl@47694
  1635
  assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
  1636
    shows "\<H>(X | Y) = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
hoelzl@47694
  1637
proof (subst conditional_entropy_generic_eq[OF _ _
hoelzl@49790
  1638
  simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@49792
  1639
  have "finite ((\<lambda>x. (X x, Y x))`space M)"
hoelzl@49792
  1640
    using XY unfolding simple_distributed_def by auto
hoelzl@49792
  1641
  from finite_imageI[OF this, of fst]
hoelzl@49792
  1642
  have [simp]: "finite (X`space M)"
hoelzl@49792
  1643
    by (simp add: image_compose[symmetric] comp_def)
hoelzl@47694
  1644
  note Y[THEN simple_distributed_finite, simp]
hoelzl@47694
  1645
  show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694
  1646
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
  1647
  show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694
  1648
    by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694
  1649
  let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
hoelzl@47694
  1650
  have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
hoelzl@47694
  1651
    (is "?P = ?C")
hoelzl@49792
  1652
    using Y by (simp add: simple_distributed_finite pair_measure_count_space)
hoelzl@47694
  1653
  have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) =
hoelzl@47694
  1654
    (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / Py (snd x)) else 0)"
hoelzl@47694
  1655
    by auto
hoelzl@49792
  1656
  from Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) =
hoelzl@47694
  1657
    - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
hoelzl@47694
  1658
    by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta')
hoelzl@47694
  1659
qed
hoelzl@39097
  1660
hoelzl@47694
  1661
lemma (in information_space) conditional_mutual_information_eq_conditional_entropy:
hoelzl@41689
  1662
  assumes X: "simple_function M X" and Y: "simple_function M Y"
hoelzl@47694
  1663
  shows "\<I>(X ; X | Y) = \<H>(X | Y)"
hoelzl@47694
  1664
proof -
hoelzl@47694
  1665
  def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0"
hoelzl@47694
  1666
  def Pxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M) else 0"
hoelzl@47694
  1667
  def Pxxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, X x, Y x))`space M then measure M ((\<lambda>x. (X x, X x, Y x)) -` {x} \<inter> space M) else 0"
hoelzl@47694
  1668
  let ?M = "X`space M \<times> X`space M \<times> Y`space M"
hoelzl@39097
  1669
hoelzl@47694
  1670
  note XY = simple_function_Pair[OF X Y]
hoelzl@47694
  1671
  note XXY = simple_function_Pair[OF X XY]
hoelzl@47694
  1672
  have Py: "simple_distributed M Y Py"
hoelzl@47694
  1673
    using Y by (rule simple_distributedI) (auto simp: Py_def)
hoelzl@47694
  1674
  have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
  1675
    using XY by (rule simple_distributedI) (auto simp: Pxy_def)
hoelzl@47694
  1676
  have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy"
hoelzl@47694
  1677
    using XXY by (rule simple_distributedI) (auto simp: Pxxy_def)
hoelzl@47694
  1678
  have eq: "(\<lambda>x. (X x, X x, Y x)) ` space M = (\<lambda>(x, y). (x, x, y)) ` (\<lambda>x. (X x, Y x)) ` space M"
hoelzl@47694
  1679
    by auto
hoelzl@47694
  1680
  have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A"
hoelzl@47694
  1681
    by (auto simp: inj_on_def)
hoelzl@47694
  1682
  have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)"
hoelzl@47694
  1683
    by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob])
hoelzl@47694
  1684
  have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
  1685
    by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair)
hoelzl@47694
  1686
  then show ?thesis
hoelzl@47694
  1687
    apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy])
hoelzl@49792
  1688
    apply (subst conditional_entropy_eq[OF Py Pxy])
hoelzl@47694
  1689
    apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj]
hoelzl@47694
  1690
                log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space)
hoelzl@47694
  1691
    using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE]
hoelzl@49790
  1692
  apply (auto simp add: not_le[symmetric] AE_count_space)
hoelzl@47694
  1693
    done
hoelzl@47694
  1694
qed
hoelzl@47694
  1695
hoelzl@47694
  1696
lemma (in information_space) conditional_entropy_nonneg:
hoelzl@47694
  1697
  assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)"
hoelzl@47694
  1698
  using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y]
hoelzl@47694
  1699
  by simp
hoelzl@36080
  1700
hoelzl@39097
  1701
subsection {* Equalities *}
hoelzl@39097
  1702
hoelzl@47694
  1703
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
hoelzl@47694
  1704
  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@47694
  1705
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@47694
  1706
  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
hoelzl@47694
  1707
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694
  1708
  assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
hoelzl@47694
  1709
  assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
hoelzl@47694
  1710
  assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
hoelzl@47694
  1711
  shows  "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@40859
  1712
proof -
hoelzl@47694
  1713
  have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@47694
  1714
    using b_gt_1 Px[THEN distributed_real_measurable]
hoelzl@49786
  1715
    apply (subst entropy_distr[OF Px])
hoelzl@47694
  1716
    apply (subst distributed_transform_integral[OF Pxy Px, where T=fst])
hoelzl@47694
  1717
    apply (auto intro!: integral_cong)
hoelzl@47694
  1718
    done
hoelzl@47694
  1719
hoelzl@47694
  1720
  have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@47694
  1721
    using b_gt_1 Py[THEN distributed_real_measurable]
hoelzl@49786
  1722
    apply (subst entropy_distr[OF Py])
hoelzl@47694
  1723
    apply (subst distributed_transform_integral[OF Pxy Py, where T=snd])
hoelzl@47694
  1724
    apply (auto intro!: integral_cong)
hoelzl@47694
  1725
    done
hoelzl@47694
  1726
hoelzl@47694
  1727
  interpret S: sigma_finite_measure S by fact
hoelzl@47694
  1728
  interpret T: sigma_finite_measure T by fact
hoelzl@47694
  1729
  interpret ST: pair_sigma_finite S T ..
hoelzl@47694
  1730
  have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" ..
hoelzl@47694
  1731
hoelzl@47694
  1732
  have XY: "entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T))"
hoelzl@49786
  1733
    by (subst entropy_distr[OF Pxy]) (auto intro!: integral_cong)
hoelzl@47694
  1734
  
hoelzl@47694
  1735
  have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
  1736
    by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair)
hoelzl@47694
  1737
  moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694
  1738
    by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
hoelzl@47694
  1739
  moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)"
hoelzl@47694
  1740
    using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
hoelzl@47694
  1741
  moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
hoelzl@47694
  1742
    using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
hoelzl@47694
  1743
  moreover note Pxy[THEN distributed_real_AE]
hoelzl@47694
  1744
  ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) = 
hoelzl@47694
  1745
    Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@47694
  1746
    (is "AE x in _. ?f x = ?g x")
hoelzl@47694
  1747
  proof eventually_elim
hoelzl@47694
  1748
    case (goal1 x)
hoelzl@47694
  1749
    show ?case
hoelzl@47694
  1750
    proof cases
hoelzl@47694
  1751
      assume "Pxy x \<noteq> 0"
hoelzl@47694
  1752
      with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x"
hoelzl@47694
  1753
        by auto
hoelzl@47694
  1754
      then show ?thesis
hoelzl@47694
  1755
        using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@47694
  1756
    qed simp
hoelzl@47694
  1757
  qed
hoelzl@47694
  1758
hoelzl@47694
  1759
  have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?f"
hoelzl@47694
  1760
    unfolding X Y XY
hoelzl@47694
  1761
    apply (subst integral_diff)
hoelzl@47694
  1762
    apply (intro integral_diff Ixy Ix Iy)+
hoelzl@47694
  1763
    apply (subst integral_diff)
hoelzl@47694
  1764
    apply (intro integral_diff Ixy Ix Iy)+
hoelzl@47694
  1765
    apply (simp add: field_simps)
hoelzl@47694
  1766
    done
hoelzl@47694
  1767
  also have "\<dots> = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?g"
hoelzl@47694
  1768
    using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE)
hoelzl@47694
  1769
  also have "\<dots> = mutual_information b S T X Y"
hoelzl@47694
  1770
    by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
hoelzl@47694
  1771
  finally show ?thesis ..
hoelzl@47694
  1772
qed
hoelzl@47694
  1773
hoelzl@49802
  1774
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy':
hoelzl@49802
  1775
  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@49802
  1776
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49802
  1777
  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
hoelzl@49802
  1778
  assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49802
  1779
  assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
hoelzl@49802
  1780
  assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
hoelzl@49802
  1781
  assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
hoelzl@49802
  1782
  shows  "mutual_information b S T X Y = entropy b S X - conditional_entropy b S T X Y"
hoelzl@49802
  1783
  using
hoelzl@49802
  1784
    mutual_information_eq_entropy_conditional_entropy_distr[OF S T Px Py Pxy Ix Iy Ixy]
hoelzl@49802
  1785
    conditional_entropy_eq_entropy[OF S T Py Pxy Ixy Iy]
hoelzl@49802
  1786
  by simp
hoelzl@49802
  1787
hoelzl@47694
  1788
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
hoelzl@47694
  1789
  assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
hoelzl@47694
  1790
  shows  "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"
hoelzl@47694
  1791
proof -
hoelzl@47694
  1792
  have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
hoelzl@47694
  1793
    using sf_X by (rule simple_distributedI) auto
hoelzl@47694
  1794
  have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
hoelzl@47694
  1795
    using sf_Y by (rule simple_distributedI) auto
hoelzl@47694
  1796
  have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694
  1797
    using sf_X sf_Y by (rule simple_function_Pair)
hoelzl@47694
  1798
  then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
hoelzl@47694
  1799
    by (rule simple_distributedI) auto
hoelzl@47694
  1800
  from simple_distributed_joint_finite[OF this, simp]
hoelzl@47694
  1801
  have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
hoelzl@47694
  1802
    by (simp add: pair_measure_count_space)
hoelzl@47694
  1803
hoelzl@47694
  1804
  have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))"
hoelzl@47694
  1805
    using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]
hoelzl@47694
  1806
    by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space)
hoelzl@47694
  1807
  then show ?thesis
hoelzl@49791
  1808
    unfolding conditional_entropy_eq_entropy_simple[OF sf_X sf_Y] by simp
hoelzl@47694
  1809
qed
hoelzl@47694
  1810
hoelzl@47694
  1811
lemma (in information_space) mutual_information_nonneg_simple:
hoelzl@47694
  1812
  assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
hoelzl@47694
  1813
  shows  "0 \<le> \<I>(X ; Y)"
hoelzl@47694
  1814
proof -
hoelzl@47694
  1815
  have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
hoelzl@47694
  1816
    using sf_X by (rule simple_distributedI) auto
hoelzl@47694
  1817
  have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
hoelzl@47694
  1818
    using sf_Y by (rule simple_distributedI) auto
hoelzl@47694
  1819
hoelzl@47694
  1820
  have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694
  1821
    using sf_X sf_Y by (rule simple_function_Pair)
hoelzl@47694
  1822
  then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
hoelzl@47694
  1823
    by (rule simple_distributedI) auto
hoelzl@47694
  1824
hoelzl@47694
  1825
  from simple_distributed_joint_finite[OF this, simp]
hoelzl@47694
  1826
  have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
hoelzl@47694
  1827
    by (simp add: pair_measure_count_space)
hoelzl@47694
  1828
hoelzl@40859
  1829
  show ?thesis
hoelzl@47694
  1830
    by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@47694
  1831
       (simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite)
hoelzl@40859
  1832
qed
hoelzl@36080
  1833
hoelzl@40859
  1834
lemma (in information_space) conditional_entropy_less_eq_entropy:
hoelzl@41689
  1835
  assumes X: "simple_function M X" and Z: "simple_function M Z"
hoelzl@40859
  1836
  shows "\<H>(X | Z) \<le> \<H>(X)"
hoelzl@36624
  1837
proof -
hoelzl@47694
  1838
  have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple)
hoelzl@47694
  1839
  also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
hoelzl@47694
  1840
  finally show ?thesis by auto
hoelzl@36080
  1841
qed
hoelzl@36080
  1842
hoelzl@49803
  1843
lemma (in information_space) 
hoelzl@49803
  1844
  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@49803
  1845
  assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49803
  1846
  assumes Px: "finite_entropy S X Px" and Py: "finite_entropy T Y Py"
hoelzl@49803
  1847
  assumes Pxy: "finite_entropy (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803
  1848
  shows "conditional_entropy b S T X Y \<le> entropy b S X"
hoelzl@49803
  1849
proof -
hoelzl@49803
  1850
hoelzl@49803
  1851
  have "0 \<le> mutual_information b S T X Y" 
hoelzl@49803
  1852
    by (rule mutual_information_nonneg') fact+
hoelzl@49803
  1853
  also have "\<dots> = entropy b S X - conditional_entropy b S T X Y"
hoelzl@49803
  1854
    apply (rule mutual_information_eq_entropy_conditional_entropy')
hoelzl@49803
  1855
    using assms
hoelzl@49803
  1856
    by (auto intro!: finite_entropy_integrable finite_entropy_distributed
hoelzl@49803
  1857
      finite_entropy_integrable_transform[OF Px]
hoelzl@49803
  1858
      finite_entropy_integrable_transform[OF Py])
hoelzl@49803
  1859
  finally show ?thesis by auto
hoelzl@49803
  1860
qed
hoelzl@49803
  1861
hoelzl@40859
  1862
lemma (in information_space) entropy_chain_rule:
hoelzl@41689
  1863
  assumes X: "simple_function M X" and Y: "simple_function M Y"
hoelzl@40859
  1864
  shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
hoelzl@40859
  1865
proof -
hoelzl@47694
  1866
  note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl]
hoelzl@47694
  1867
  note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl]
hoelzl@47694
  1868
  note simple_distributed_joint_finite[OF this, simp]
hoelzl@47694
  1869
  let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)"
hoelzl@47694
  1870
  let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)"
hoelzl@47694
  1871
  let ?h = "\<lambda>x. if x \<in> (\<lambda>x. (Y x, X x)) ` space M then prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M) else 0"
hoelzl@47694
  1872
  have "\<H>(\<lambda>x. (X x, Y x)) = - (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` space M. ?f x * log b (?f x))"
hoelzl@47694
  1873
    using XY by (rule entropy_simple_distributed)
hoelzl@47694
  1874
  also have "\<dots> = - (\<Sum>x\<in>(\<lambda>(x, y). (y, x)) ` (\<lambda>x. (X x, Y x)) ` space M. ?g x * log b (?g x))"
hoelzl@47694
  1875
    by (subst (2) setsum_reindex) (auto simp: inj_on_def intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
hoelzl@47694
  1876
  also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))"
hoelzl@47694
  1877
    by (auto intro!: setsum_cong)
hoelzl@47694
  1878
  also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))"
hoelzl@49786
  1879
    by (subst entropy_distr[OF simple_distributed_joint[OF YX]])
hoelzl@47694
  1880
       (auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite
hoelzl@47694
  1881
             cong del: setsum_cong  intro!: setsum_mono_zero_left)
hoelzl@47694
  1882
  finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" .
hoelzl@47694
  1883
  then show ?thesis
hoelzl@49791
  1884
    unfolding conditional_entropy_eq_entropy_simple[OF Y X] by simp
hoelzl@36624
  1885
qed
hoelzl@36624
  1886
hoelzl@40859
  1887
lemma (in information_space) entropy_partition:
hoelzl@47694
  1888
  assumes X: "simple_function M X"
hoelzl@47694
  1889
  shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)"
hoelzl@36624
  1890
proof -
hoelzl@47694
  1891
  note fX = simple_function_compose[OF X, of f]  
hoelzl@47694
  1892
  have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto
hoelzl@47694
  1893
  have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A"
hoelzl@47694
  1894
    by (auto simp: inj_on_def)
hoelzl@47694
  1895
  show ?thesis
hoelzl@47694
  1896
    apply (subst entropy_chain_rule[symmetric, OF fX X])
hoelzl@47694
  1897
    apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]])
hoelzl@47694
  1898
    apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
hoelzl@47694
  1899
    unfolding eq
hoelzl@47694
  1900
    apply (subst setsum_reindex[OF inj])
hoelzl@47694
  1901
    apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
hoelzl@47694
  1902
    done
hoelzl@36624
  1903
qed
hoelzl@36624
  1904
hoelzl@40859
  1905
corollary (in information_space) entropy_data_processing:
hoelzl@41689
  1906
  assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
hoelzl@40859
  1907
proof -
hoelzl@47694
  1908
  note fX = simple_function_compose[OF X, of f]
hoelzl@47694
  1909
  from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
hoelzl@40859
  1910
  then show "\<H>(f \<circ> X) \<le> \<H>(X)"
hoelzl@47694
  1911
    by (auto intro: conditional_entropy_nonneg[OF X fX])
hoelzl@40859
  1912
qed
hoelzl@36624
  1913
hoelzl@40859
  1914
corollary (in information_space) entropy_of_inj:
hoelzl@41689
  1915
  assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
hoelzl@36624
  1916
  shows "\<H>(f \<circ> X) = \<H>(X)"
hoelzl@36624
  1917
proof (rule antisym)
hoelzl@40859
  1918
  show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
hoelzl@36624
  1919
next
hoelzl@41689
  1920
  have sf: "simple_function M (f \<circ> X)"
hoelzl@40859
  1921
    using X by auto
hoelzl@36624
  1922
  have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
hoelzl@47694
  1923
    unfolding o_assoc
hoelzl@47694
  1924
    apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
hoelzl@47694
  1925
    apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"])
hoelzl@47694
  1926
    apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def)
hoelzl@47694
  1927
    done
hoelzl@36624
  1928
  also have "... \<le> \<H>(f \<circ> X)"
hoelzl@40859
  1929
    using entropy_data_processing[OF sf] .
hoelzl@36624
  1930
  finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
hoelzl@36624
  1931
qed
hoelzl@36624
  1932
hoelzl@36080
  1933
end