
hoelzl@42067: (*  Title:      HOL/Probability/Information.thy
hoelzl@42067:     Author:     Johannes Hölzl, TU München
hoelzl@42067:     Author:     Armin Heller, TU München
hoelzl@42067: *)
hoelzl@42067: 
hoelzl@42067: header {*Information theory*}
hoelzl@42067: 
hoelzl@36080: theory Information
wenzelm@41413: imports
hoelzl@43340:   Independent_Family
hoelzl@43556:   Radon_Nikodym
wenzelm@41413:   "~~/src/HOL/Library/Convex"
hoelzl@36080: begin
hoelzl@36080: 
hoelzl@39097: lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
hoelzl@39097:   by (subst log_le_cancel_iff) auto
hoelzl@39097: 
hoelzl@39097: lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
hoelzl@39097:   by (subst log_less_cancel_iff) auto
hoelzl@39097: 
hoelzl@39097: lemma setsum_cartesian_product':
hoelzl@39097:   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
hoelzl@39097:   unfolding setsum_cartesian_product by simp
hoelzl@39097: 
hoelzl@39097: lemma split_pairs:
hoelzl@40859:   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
hoelzl@40859:   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
hoelzl@38656: 
hoelzl@38656: section "Information theory"
hoelzl@38656: 
hoelzl@40859: locale information_space = prob_space +
hoelzl@38656:   fixes b :: real assumes b_gt_1: "1 < b"
hoelzl@38656: 
hoelzl@40859: context information_space
hoelzl@38656: begin
hoelzl@38656: 
hoelzl@40859: text {* Introduce some simplification rules for logarithm of base @{term b}. *}
hoelzl@40859: 
hoelzl@40859: lemma log_neg_const:
hoelzl@40859:   assumes "x \<le> 0"
hoelzl@40859:   shows "log b x = log b 0"
hoelzl@36624: proof -
hoelzl@40859:   { fix u :: real
hoelzl@40859:     have "x \<le> 0" by fact
hoelzl@40859:     also have "0 < exp u"
hoelzl@40859:       using exp_gt_zero .
hoelzl@40859:     finally have "exp u \<noteq> x"
hoelzl@40859:       by auto }
hoelzl@40859:   then show "log b x = log b 0"
hoelzl@40859:     by (simp add: log_def ln_def)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@40859: lemma log_mult_eq:
hoelzl@40859:   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
hoelzl@40859:   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
hoelzl@40859:   by (auto simp: zero_less_mult_iff mult_le_0_iff)
hoelzl@38656: 
hoelzl@40859: lemma log_inverse_eq:
hoelzl@40859:   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
hoelzl@40859:   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
hoelzl@36080: 
hoelzl@40859: lemma log_divide_eq:
hoelzl@40859:   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
hoelzl@40859:   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
hoelzl@40859:   by (auto simp: zero_less_mult_iff mult_le_0_iff)
hoelzl@38656: 
hoelzl@40859: lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
hoelzl@38656: 
hoelzl@38656: end
hoelzl@38656: 
hoelzl@39097: subsection "Kullback$-$Leibler divergence"
hoelzl@36080: 
hoelzl@39097: text {* The Kullback$-$Leibler divergence is also known as relative entropy or
hoelzl@39097: Kullback$-$Leibler distance. *}
hoelzl@39097: 
hoelzl@39097: definition
hoelzl@47694:   "entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N"
hoelzl@43340: 
hoelzl@43340: definition
wenzelm@53015:   "KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)"
hoelzl@43340: 
hoelzl@43340: lemma (in information_space) measurable_entropy_density:
hoelzl@47694:   assumes ac: "absolutely_continuous M N" "sets N = events"
hoelzl@47694:   shows "entropy_density b M N \<in> borel_measurable M"
hoelzl@43340: proof -
hoelzl@47694:   from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis
hoelzl@50003:     unfolding entropy_density_def by auto
hoelzl@43340: qed
hoelzl@43340: 
hoelzl@50003: lemma borel_measurable_RN_deriv_density[measurable (raw)]:
hoelzl@50003:   "f \<in> borel_measurable M \<Longrightarrow> RN_deriv M (density M f) \<in> borel_measurable M"
hoelzl@50003:   using borel_measurable_RN_deriv_density[of "\<lambda>x. max 0 (f x )" M]
hoelzl@50003:   by (simp add: density_max_0[symmetric])
hoelzl@50003: 
hoelzl@47694: lemma (in sigma_finite_measure) KL_density:
hoelzl@47694:   fixes f :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "1 < b"
hoelzl@47694:   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
hoelzl@47694:   shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)"
hoelzl@47694:   unfolding KL_divergence_def
hoelzl@47694: proof (subst integral_density)
hoelzl@47694:   show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
hoelzl@49776:     using f
hoelzl@50003:     by (auto simp: comp_def entropy_density_def)
hoelzl@47694:   have "density M (RN_deriv M (density M f)) = density M f"
hoelzl@47694:     using f by (intro density_RN_deriv_density) auto
hoelzl@47694:   then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
hoelzl@47694:     using f
hoelzl@47694:     by (intro density_unique)
hoelzl@47694:        (auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg)
hoelzl@47694:   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)"
hoelzl@47694:     apply (intro integral_cong_AE)
hoelzl@47694:     using eq
hoelzl@47694:     apply eventually_elim
hoelzl@47694:     apply (auto simp: entropy_density_def)
hoelzl@47694:     done
hoelzl@47694: qed fact+
hoelzl@43340: 
hoelzl@47694: lemma (in sigma_finite_measure) KL_density_density:
hoelzl@47694:   fixes f g :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "1 < b"
hoelzl@47694:   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
hoelzl@47694:   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
hoelzl@47694:   assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
hoelzl@47694:   shows "KL_divergence b (density M f) (density M g) = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
hoelzl@47694: proof -
hoelzl@47694:   interpret Mf: sigma_finite_measure "density M f"
hoelzl@47694:     using f by (subst sigma_finite_iff_density_finite) auto
hoelzl@47694:   have "KL_divergence b (density M f) (density M g) =
hoelzl@47694:     KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
hoelzl@47694:     using f g ac by (subst density_density_divide) simp_all
hoelzl@47694:   also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)"
hoelzl@47694:     using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density divide_nonneg_nonneg)
hoelzl@47694:   also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
hoelzl@47694:     using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE)
hoelzl@47694:   finally show ?thesis .
hoelzl@47694: qed
hoelzl@43340: 
hoelzl@47694: lemma (in information_space) KL_gt_0:
hoelzl@47694:   fixes D :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "prob_space (density M D)"
hoelzl@47694:   assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
hoelzl@47694:   assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694:   assumes A: "density M D \<noteq> M"
hoelzl@47694:   shows "0 < KL_divergence b M (density M D)"
hoelzl@47694: proof -
hoelzl@47694:   interpret N: prob_space "density M D" by fact
hoelzl@43340: 
hoelzl@47694:   obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A"
hoelzl@47694:     using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto
hoelzl@47694: 
hoelzl@47694:   let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
hoelzl@47694:   have [simp, intro]: "?D_set \<in> sets M"
hoelzl@47694:     using D by auto
hoelzl@43340: 
wenzelm@53015:   have D_neg: "(\<integral>\<^sup>+ x. ereal (- D x) \<partial>M) = 0"
hoelzl@43340:     using D by (subst positive_integral_0_iff_AE) auto
hoelzl@43340: 
wenzelm@53015:   have "(\<integral>\<^sup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)"
hoelzl@47694:     using D by (simp add: emeasure_density cong: positive_integral_cong)
wenzelm@53015:   then have D_pos: "(\<integral>\<^sup>+ x. ereal (D x) \<partial>M) = 1"
hoelzl@47694:     using N.emeasure_space_1 by simp
hoelzl@43340: 
wenzelm@53015:   have "integrable M D" "integral\<^sup>L M D = 1"
hoelzl@47694:     using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all
hoelzl@43340: 
hoelzl@47694:   have "0 \<le> 1 - measure M ?D_set"
hoelzl@43340:     using prob_le_1 by (auto simp: field_simps)
hoelzl@43340:   also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
wenzelm@53015:     using `integrable M D` `integral\<^sup>L M D = 1`
hoelzl@47694:     by (simp add: emeasure_eq_measure)
hoelzl@47694:   also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
hoelzl@43340:   proof (rule integral_less_AE)
hoelzl@43340:     show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
hoelzl@43340:       using `integrable M D`
hoelzl@43340:       by (intro integral_diff integral_indicator) auto
hoelzl@43340:   next
hoelzl@47694:     from integral_cmult(1)[OF int, of "ln b"]
hoelzl@47694:     show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" 
hoelzl@47694:       by (simp add: ac_simps)
hoelzl@43340:   next
hoelzl@47694:     show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
hoelzl@43340:     proof
hoelzl@47694:       assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
hoelzl@47694:       then have disj: "AE x in M. D x = 1 \<or> D x = 0"
immler@50244:         using D(1) by (auto intro!: AE_I[OF subset_refl] sets.sets_Collect)
hoelzl@43340: 
wenzelm@53015:       have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^sup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
hoelzl@43340:         using D(1) by auto
wenzelm@53015:       also have "\<dots> = (\<integral>\<^sup>+ x. ereal (D x) \<partial>M)"
hoelzl@43920:         using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
hoelzl@47694:       finally have "AE x in M. D x = 1"
hoelzl@47694:         using D D_pos by (intro AE_I_eq_1) auto
wenzelm@53015:       then have "(\<integral>\<^sup>+x. indicator A x\<partial>M) = (\<integral>\<^sup>+x. ereal (D x) * indicator A x\<partial>M)"
hoelzl@43920:         by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
hoelzl@47694:       also have "\<dots> = density M D A"
hoelzl@47694:         using `A \<in> sets M` D by (simp add: emeasure_density)
hoelzl@47694:       finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp
hoelzl@43340:     qed
hoelzl@43340:     show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
immler@50244:       using D(1) by (auto intro: sets.sets_Collect_conj)
hoelzl@43340: 
hoelzl@47694:     show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
hoelzl@47694:       D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))"
hoelzl@43340:       using D(2)
hoelzl@47694:     proof (eventually_elim, safe)
hoelzl@47694:       fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t"
hoelzl@47694:         and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))"
hoelzl@43340: 
hoelzl@43340:       have "D t - 1 = D t - indicator ?D_set t"
hoelzl@43340:         using Dt by simp
hoelzl@43340:       also note eq
hoelzl@47694:       also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
hoelzl@47694:         using b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
hoelzl@47694:         by (simp add: log_def ln_div less_le)
hoelzl@43340:       finally have "ln (1 / D t) = 1 / D t - 1"
hoelzl@43340:         using `D t \<noteq> 0` by (auto simp: field_simps)
hoelzl@43340:       from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
hoelzl@43340:       show False by auto
hoelzl@43340:     qed
hoelzl@43340: 
hoelzl@47694:     show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
hoelzl@47694:       using D(2) AE_space
hoelzl@47694:     proof eventually_elim
hoelzl@47694:       fix t assume "t \<in> space M" "0 \<le> D t"
hoelzl@47694:       show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
hoelzl@43340:       proof cases
hoelzl@43340:         assume asm: "D t \<noteq> 0"
hoelzl@43340:         then have "0 < D t" using `0 \<le> D t` by auto
hoelzl@43340:         then have "0 < 1 / D t" by auto
hoelzl@43340:         have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
hoelzl@43340:           using asm `t \<in> space M` by (simp add: field_simps)
hoelzl@43340:         also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
hoelzl@43340:           using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
hoelzl@47694:         also have "\<dots> = D t * (ln b * log b (D t))"
hoelzl@47694:           using `0 < D t` b_gt_1
hoelzl@47694:           by (simp_all add: log_def ln_div)
hoelzl@43340:         finally show ?thesis by simp
hoelzl@43340:       qed simp
hoelzl@43340:     qed
hoelzl@43340:   qed
hoelzl@47694:   also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)"
hoelzl@47694:     by (simp add: ac_simps)
hoelzl@47694:   also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)"
hoelzl@47694:     using int by (rule integral_cmult)
hoelzl@47694:   finally show ?thesis
hoelzl@47694:     using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
hoelzl@43340: qed
hoelzl@43340: 
hoelzl@47694: lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0"
hoelzl@43340: proof -
hoelzl@47694:   have "AE x in M. 1 = RN_deriv M M x"
hoelzl@43340:   proof (rule RN_deriv_unique)
hoelzl@47694:     show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto
hoelzl@47694:     show "density M (\<lambda>x. 1) = M"
hoelzl@47694:       apply (auto intro!: measure_eqI emeasure_density)
hoelzl@47694:       apply (subst emeasure_density)
hoelzl@47694:       apply auto
hoelzl@47694:       done
hoelzl@43340:   qed
hoelzl@47694:   then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
hoelzl@43340:     by (elim AE_mp) simp
hoelzl@43340:   from integral_cong_AE[OF this]
wenzelm@53015:   have "integral\<^sup>L M (entropy_density b M M) = 0"
hoelzl@43340:     by (simp add: entropy_density_def comp_def)
hoelzl@47694:   then show "KL_divergence b M M = 0"
hoelzl@43340:     unfolding KL_divergence_def
hoelzl@47694:     by auto
hoelzl@43340: qed
hoelzl@43340: 
hoelzl@47694: lemma (in information_space) KL_eq_0_iff_eq:
hoelzl@47694:   fixes D :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "prob_space (density M D)"
hoelzl@47694:   assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
hoelzl@47694:   assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694:   shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M"
hoelzl@47694:   using KL_same_eq_0[of b] KL_gt_0[OF assms]
hoelzl@47694:   by (auto simp: less_le)
hoelzl@43340: 
hoelzl@47694: lemma (in information_space) KL_eq_0_iff_eq_ac:
hoelzl@47694:   fixes D :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "prob_space N"
hoelzl@47694:   assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694:   assumes int: "integrable N (entropy_density b M N)"
hoelzl@47694:   shows "KL_divergence b M N = 0 \<longleftrightarrow> N = M"
hoelzl@41833: proof -
hoelzl@47694:   interpret N: prob_space N by fact
hoelzl@47694:   have "finite_measure N" by unfold_locales
hoelzl@47694:   from real_RN_deriv[OF this ac] guess D . note D = this
hoelzl@47694:   
hoelzl@47694:   have "N = density M (RN_deriv M N)"
hoelzl@47694:     using ac by (rule density_RN_deriv[symmetric])
hoelzl@47694:   also have "\<dots> = density M D"
hoelzl@47694:     using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong)
hoelzl@47694:   finally have N: "N = density M D" .
hoelzl@41833: 
hoelzl@47694:   from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
hoelzl@47694:   have "integrable N (\<lambda>x. log b (D x))"
hoelzl@47694:     by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
hoelzl@47694:        (auto simp: N entropy_density_def)
hoelzl@47694:   with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694:     by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def)
hoelzl@47694:   with `prob_space N` D show ?thesis
hoelzl@47694:     unfolding N
hoelzl@47694:     by (intro KL_eq_0_iff_eq) auto
hoelzl@41833: qed
hoelzl@41833: 
hoelzl@47694: lemma (in information_space) KL_nonneg:
hoelzl@47694:   assumes "prob_space (density M D)"
hoelzl@47694:   assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
hoelzl@47694:   assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
hoelzl@47694:   shows "0 \<le> KL_divergence b M (density M D)"
hoelzl@47694:   using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0)
hoelzl@40859: 
hoelzl@47694: lemma (in sigma_finite_measure) KL_density_density_nonneg:
hoelzl@47694:   fixes f g :: "'a \<Rightarrow> real"
hoelzl@47694:   assumes "1 < b"
hoelzl@47694:   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)"
hoelzl@47694:   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)"
hoelzl@47694:   assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
hoelzl@47694:   assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))"
hoelzl@47694:   shows "0 \<le> KL_divergence b (density M f) (density M g)"
hoelzl@47694: proof -
hoelzl@47694:   interpret Mf: prob_space "density M f" by fact
hoelzl@47694:   interpret Mf: information_space "density M f" b by default fact
hoelzl@47694:   have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _")
hoelzl@47694:     using f g ac by (subst density_density_divide) simp_all
hoelzl@36080: 
hoelzl@47694:   have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
hoelzl@47694:   proof (rule Mf.KL_nonneg)
hoelzl@47694:     show "prob_space ?DD" unfolding eq by fact
hoelzl@47694:     from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)"
hoelzl@47694:       by auto
hoelzl@47694:     show "AE x in density M f. 0 \<le> g x / f x"
hoelzl@47694:       using f g by (auto simp: AE_density divide_nonneg_nonneg)
hoelzl@47694:     show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
hoelzl@47694:       using `1 < b` f g ac
hoelzl@47694:       by (subst integral_density)
hoelzl@47694:          (auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
hoelzl@47694:   qed
hoelzl@47694:   also have "\<dots> = KL_divergence b (density M f) (density M g)"
hoelzl@47694:     using f g ac by (subst density_density_divide) simp_all
hoelzl@47694:   finally show ?thesis .
hoelzl@36080: qed
hoelzl@36080: 
hoelzl@49803: subsection {* Finite Entropy *}
hoelzl@49803: 
hoelzl@49803: definition (in information_space) 
hoelzl@49803:   "finite_entropy S X f \<longleftrightarrow> distributed M S X f \<and> integrable S (\<lambda>x. f x * log b (f x))"
hoelzl@49803: 
hoelzl@49803: lemma (in information_space) finite_entropy_simple_function:
hoelzl@49803:   assumes X: "simple_function M X"
hoelzl@49803:   shows "finite_entropy (count_space (X`space M)) X (\<lambda>a. measure M {x \<in> space M. X x = a})"
hoelzl@49803:   unfolding finite_entropy_def
hoelzl@49803: proof
hoelzl@49803:   have [simp]: "finite (X ` space M)"
hoelzl@49803:     using X by (auto simp: simple_function_def)
hoelzl@49803:   then show "integrable (count_space (X ` space M))
hoelzl@49803:      (\<lambda>x. prob {xa \<in> space M. X xa = x} * log b (prob {xa \<in> space M. X xa = x}))"
hoelzl@49803:     by (rule integrable_count_space)
hoelzl@49803:   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:     by (rule distributed_simple_function_superset[OF X]) (auto intro!: arg_cong[where f=prob])
hoelzl@49803:   show "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (prob {xa \<in> space M. X xa = x}))"
hoelzl@49803:     by (rule distributed_cong_density[THEN iffD1, OF _ _ _ d]) auto
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@49803: lemma distributed_transform_AE:
hoelzl@49803:   assumes T: "T \<in> measurable P Q" "absolutely_continuous Q (distr P Q T)"
hoelzl@49803:   assumes g: "distributed M Q Y g"
hoelzl@49803:   shows "AE x in P. 0 \<le> g (T x)"
hoelzl@49803:   using g
hoelzl@49803:   apply (subst AE_distr_iff[symmetric, OF T(1)])
hoelzl@50003:   apply simp
hoelzl@49803:   apply (rule absolutely_continuous_AE[OF _ T(2)])
hoelzl@49803:   apply simp
hoelzl@49803:   apply (simp add: distributed_AE)
hoelzl@49803:   done
hoelzl@49803: 
hoelzl@49803: lemma ac_fst:
hoelzl@49803:   assumes "sigma_finite_measure T"
wenzelm@53015:   shows "absolutely_continuous S (distr (S \<Otimes>\<^sub>M T) S fst)"
hoelzl@49803: proof -
hoelzl@49803:   interpret sigma_finite_measure T by fact
hoelzl@49803:   { fix A assume "A \<in> sets S" "emeasure S A = 0"
wenzelm@53015:     moreover then have "fst -` A \<inter> space (S \<Otimes>\<^sub>M T) = A \<times> space T"
immler@50244:       by (auto simp: space_pair_measure dest!: sets.sets_into_space)
wenzelm@53015:     ultimately have "emeasure (S \<Otimes>\<^sub>M T) (fst -` A \<inter> space (S \<Otimes>\<^sub>M T)) = 0"
hoelzl@49803:       by (simp add: emeasure_pair_measure_Times) }
hoelzl@49803:   then show ?thesis
hoelzl@49803:     unfolding absolutely_continuous_def
hoelzl@49803:     apply (auto simp: null_sets_distr_iff)
hoelzl@49803:     apply (auto simp: null_sets_def intro!: measurable_sets)
hoelzl@49803:     done
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@49803: lemma ac_snd:
hoelzl@49803:   assumes "sigma_finite_measure T"
wenzelm@53015:   shows "absolutely_continuous T (distr (S \<Otimes>\<^sub>M T) T snd)"
hoelzl@49803: proof -
hoelzl@49803:   interpret sigma_finite_measure T by fact
hoelzl@49803:   { fix A assume "A \<in> sets T" "emeasure T A = 0"
wenzelm@53015:     moreover then have "snd -` A \<inter> space (S \<Otimes>\<^sub>M T) = space S \<times> A"
immler@50244:       by (auto simp: space_pair_measure dest!: sets.sets_into_space)
wenzelm@53015:     ultimately have "emeasure (S \<Otimes>\<^sub>M T) (snd -` A \<inter> space (S \<Otimes>\<^sub>M T)) = 0"
hoelzl@49803:       by (simp add: emeasure_pair_measure_Times) }
hoelzl@49803:   then show ?thesis
hoelzl@49803:     unfolding absolutely_continuous_def
hoelzl@49803:     apply (auto simp: null_sets_distr_iff)
hoelzl@49803:     apply (auto simp: null_sets_def intro!: measurable_sets)
hoelzl@49803:     done
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@49803: lemma distributed_integrable:
hoelzl@49803:   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow>
hoelzl@49803:     integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
hoelzl@50003:   by (auto simp: distributed_real_AE
hoelzl@49803:                     distributed_distr_eq_density[symmetric] integral_density[symmetric] integrable_distr_eq)
hoelzl@49803:   
hoelzl@49803: lemma distributed_transform_integrable:
hoelzl@49803:   assumes Px: "distributed M N X Px"
hoelzl@49803:   assumes "distributed M P Y Py"
hoelzl@49803:   assumes Y: "Y = (\<lambda>x. T (X x))" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
hoelzl@49803:   shows "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@49803: proof -
hoelzl@49803:   have "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable M (\<lambda>x. f (Y x))"
hoelzl@49803:     by (rule distributed_integrable) fact+
hoelzl@49803:   also have "\<dots> \<longleftrightarrow> integrable M (\<lambda>x. f (T (X x)))"
hoelzl@49803:     using Y by simp
hoelzl@49803:   also have "\<dots> \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
hoelzl@49803:     using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
hoelzl@49803:   finally show ?thesis .
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@49803: 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:   using integrable_cong_AE by blast
hoelzl@49803: 
hoelzl@49803: lemma (in information_space) finite_entropy_integrable:
hoelzl@49803:   "finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))"
hoelzl@49803:   unfolding finite_entropy_def by auto
hoelzl@49803: 
hoelzl@49803: lemma (in information_space) finite_entropy_distributed:
hoelzl@49803:   "finite_entropy S X Px \<Longrightarrow> distributed M S X Px"
hoelzl@49803:   unfolding finite_entropy_def by auto
hoelzl@49803: 
hoelzl@49803: lemma (in information_space) finite_entropy_integrable_transform:
hoelzl@49803:   assumes Fx: "finite_entropy S X Px"
hoelzl@49803:   assumes Fy: "distributed M T Y Py"
hoelzl@49803:     and "X = (\<lambda>x. f (Y x))"
hoelzl@49803:     and "f \<in> measurable T S"
hoelzl@49803:   shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))"
hoelzl@49803:   using assms unfolding finite_entropy_def
hoelzl@49803:   using distributed_transform_integrable[of M T Y Py S X Px f "\<lambda>x. log b (Px x)"]
hoelzl@50003:   by auto
hoelzl@49803: 
hoelzl@39097: subsection {* Mutual Information *}
hoelzl@39097: 
hoelzl@36080: definition (in prob_space)
hoelzl@38656:   "mutual_information b S T X Y =
wenzelm@53015:     KL_divergence b (distr M S X \<Otimes>\<^sub>M distr M T Y) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
hoelzl@36080: 
hoelzl@47694: lemma (in information_space) mutual_information_indep_vars:
hoelzl@43340:   fixes S T X Y
wenzelm@53015:   defines "P \<equiv> distr M S X \<Otimes>\<^sub>M distr M T Y"
wenzelm@53015:   defines "Q \<equiv> distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@43340:   shows "indep_var S X T Y \<longleftrightarrow>
hoelzl@43340:     (random_variable S X \<and> random_variable T Y \<and>
hoelzl@47694:       absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and>
hoelzl@47694:       mutual_information b S T X Y = 0)"
hoelzl@47694:   unfolding indep_var_distribution_eq
hoelzl@43340: proof safe
hoelzl@50003:   assume rv[measurable]: "random_variable S X" "random_variable T Y"
hoelzl@43340: 
hoelzl@47694:   interpret X: prob_space "distr M S X"
hoelzl@47694:     by (rule prob_space_distr) fact
hoelzl@47694:   interpret Y: prob_space "distr M T Y"
hoelzl@47694:     by (rule prob_space_distr) fact
hoelzl@47694:   interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default
hoelzl@47694:   interpret P: information_space P b unfolding P_def by default (rule b_gt_1)
hoelzl@43340: 
hoelzl@47694:   interpret Q: prob_space Q unfolding Q_def
hoelzl@50003:     by (rule prob_space_distr) simp
hoelzl@43340: 
wenzelm@53015:   { assume "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694:     then have [simp]: "Q = P"  unfolding Q_def P_def by simp
hoelzl@43340: 
hoelzl@47694:     show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
hoelzl@47694:     then have ed: "entropy_density b P Q \<in> borel_measurable P"
hoelzl@47694:       by (rule P.measurable_entropy_density) simp
hoelzl@43340: 
hoelzl@47694:     have "AE x in P. 1 = RN_deriv P Q x"
hoelzl@47694:     proof (rule P.RN_deriv_unique)
hoelzl@47694:       show "density P (\<lambda>x. 1) = Q"
hoelzl@47694:         unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density)
hoelzl@47694:     qed auto
hoelzl@47694:     then have ae_0: "AE x in P. entropy_density b P Q x = 0"
hoelzl@47694:       by eventually_elim (auto simp: entropy_density_def)
hoelzl@47694:     then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)"
hoelzl@47694:       using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
hoelzl@47694:     then show "integrable Q (entropy_density b P Q)" by simp
hoelzl@43340: 
hoelzl@47694:     show "mutual_information b S T X Y = 0"
hoelzl@47694:       unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
hoelzl@47694:       using ae_0 by (simp cong: integral_cong_AE) }
hoelzl@43340: 
hoelzl@47694:   { assume ac: "absolutely_continuous P Q"
hoelzl@47694:     assume int: "integrable Q (entropy_density b P Q)"
hoelzl@47694:     assume I_eq_0: "mutual_information b S T X Y = 0"
hoelzl@43340: 
hoelzl@47694:     have eq: "Q = P"
hoelzl@47694:     proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1])
hoelzl@47694:       show "prob_space Q" by unfold_locales
hoelzl@47694:       show "absolutely_continuous P Q" by fact
hoelzl@47694:       show "integrable Q (entropy_density b P Q)" by fact
hoelzl@47694:       show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure)
hoelzl@47694:       show "KL_divergence b P Q = 0"
hoelzl@47694:         using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def)
hoelzl@47694:     qed
wenzelm@53015:     then show "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@47694:       unfolding P_def Q_def .. }
hoelzl@43340: qed
hoelzl@43340: 
hoelzl@40859: abbreviation (in information_space)
hoelzl@40859:   mutual_information_Pow ("\<I>'(_ ; _')") where
hoelzl@47694:   "\<I>(X ; Y) \<equiv> mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y"
hoelzl@41689: 
hoelzl@47694: lemma (in information_space)
hoelzl@47694:   fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@49803:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49803:   assumes Fx: "finite_entropy S X Px" and Fy: "finite_entropy T Y Py"
wenzelm@53015:   assumes Fxy: "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803:   defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
wenzelm@53015:   shows mutual_information_distr': "mutual_information b S T X Y = integral\<^sup>L (S \<Otimes>\<^sub>M T) f" (is "?M = ?R")
hoelzl@49803:     and mutual_information_nonneg': "0 \<le> mutual_information b S T X Y"
hoelzl@49803: proof -
hoelzl@49803:   have Px: "distributed M S X Px"
hoelzl@49803:     using Fx by (auto simp: finite_entropy_def)
hoelzl@49803:   have Py: "distributed M T Y Py"
hoelzl@49803:     using Fy by (auto simp: finite_entropy_def)
wenzelm@53015:   have Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803:     using Fxy by (auto simp: finite_entropy_def)
hoelzl@49803: 
hoelzl@49803:   have X: "random_variable S X"
hoelzl@50003:     using Px by auto
hoelzl@49803:   have Y: "random_variable T Y"
hoelzl@50003:     using Py by auto
hoelzl@49803:   interpret S: sigma_finite_measure S by fact
hoelzl@49803:   interpret T: sigma_finite_measure T by fact
hoelzl@49803:   interpret ST: pair_sigma_finite S T ..
hoelzl@49803:   interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
hoelzl@49803:   interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
hoelzl@49803:   interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
wenzelm@53015:   let ?P = "S \<Otimes>\<^sub>M T"
hoelzl@49803:   let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"
hoelzl@49803: 
hoelzl@49803:   { fix A assume "A \<in> sets S"
hoelzl@49803:     with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
hoelzl@49803:       by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@49803:                intro!: arg_cong[where f="emeasure M"]) }
hoelzl@49803:   note marginal_eq1 = this
hoelzl@49803:   { fix A assume "A \<in> sets T"
hoelzl@49803:     with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
hoelzl@49803:       by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@49803:                intro!: arg_cong[where f="emeasure M"]) }
hoelzl@49803:   note marginal_eq2 = this
hoelzl@49803: 
hoelzl@49803:   have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
hoelzl@49803:     by auto
hoelzl@49803: 
wenzelm@53015:   have distr_eq: "distr M S X \<Otimes>\<^sub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
hoelzl@49803:     unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
hoelzl@49803:   proof (subst pair_measure_density)
hoelzl@49803:     show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
hoelzl@49803:       "AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
hoelzl@49803:       using Px Py by (auto simp: distributed_def)
hoelzl@49803:     show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
hoelzl@49803:   qed (fact | simp)+
hoelzl@49803:   
hoelzl@49803:   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:     unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..
hoelzl@49803: 
hoelzl@49803:   from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
hoelzl@49803:     by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
hoelzl@49803:   have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
hoelzl@49803:   proof (rule ST.AE_pair_measure)
hoelzl@49803:     show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
hoelzl@49803:       using f by auto
hoelzl@49803:     show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
hoelzl@49803:       using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
hoelzl@49803:   qed
hoelzl@49803: 
hoelzl@49803:   have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@49803:     by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@49803:   moreover
hoelzl@49803:   have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@49803:     by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@49803:   ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@49803:     by eventually_elim auto
hoelzl@49803: 
hoelzl@49803:   show "?M = ?R"
hoelzl@49803:     unfolding M f_def
hoelzl@49803:     using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
hoelzl@49803:     by (rule ST.KL_density_density)
hoelzl@49803: 
hoelzl@49803:   have X: "X = fst \<circ> (\<lambda>x. (X x, Y x))" and Y: "Y = snd \<circ> (\<lambda>x. (X x, Y x))"
hoelzl@49803:     by auto
hoelzl@49803: 
wenzelm@53015:   have "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)))"
hoelzl@49803:     using finite_entropy_integrable[OF Fxy]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fx Pxy, of fst]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fy Pxy, of snd]
hoelzl@49803:     by simp
wenzelm@53015:   moreover have "f \<in> borel_measurable (S \<Otimes>\<^sub>M T)"
hoelzl@49803:     unfolding f_def using Px Py Pxy
hoelzl@49803:     by (auto intro: distributed_real_measurable measurable_fst'' measurable_snd''
hoelzl@49803:       intro!: borel_measurable_times borel_measurable_log borel_measurable_divide)
wenzelm@53015:   ultimately have int: "integrable (S \<Otimes>\<^sub>M T) f"
hoelzl@49803:     apply (rule integrable_cong_AE_imp)
hoelzl@49803:     using
hoelzl@49803:       distributed_transform_AE[OF measurable_fst ac_fst, of T, OF T Px]
hoelzl@49803:       distributed_transform_AE[OF measurable_snd ac_snd, of _ _ _ _ S, OF T Py]
hoelzl@49803:       subdensity_real[OF measurable_fst Pxy Px X]
hoelzl@49803:       subdensity_real[OF measurable_snd Pxy Py Y]
hoelzl@49803:       distributed_real_AE[OF Pxy]
hoelzl@49803:     by eventually_elim
hoelzl@49803:        (auto simp: f_def log_divide_eq log_mult_eq field_simps zero_less_mult_iff mult_nonneg_nonneg)
hoelzl@49803: 
hoelzl@49803:   show "0 \<le> ?M" unfolding M
hoelzl@49803:   proof (rule ST.KL_density_density_nonneg
hoelzl@49803:     [OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
wenzelm@53015:     show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x))) "
hoelzl@49803:       unfolding distributed_distr_eq_density[OF Pxy, symmetric]
hoelzl@49803:       using distributed_measurable[OF Pxy] by (rule prob_space_distr)
wenzelm@53015:     show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
hoelzl@49803:       unfolding distr_eq[symmetric] by unfold_locales
hoelzl@49803:   qed
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@49803: 
hoelzl@49803: lemma (in information_space)
hoelzl@49803:   fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694:   assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@47694:   assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:   defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
wenzelm@53015:   shows mutual_information_distr: "mutual_information b S T X Y = integral\<^sup>L (S \<Otimes>\<^sub>M T) f" (is "?M = ?R")
wenzelm@53015:     and mutual_information_nonneg: "integrable (S \<Otimes>\<^sub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y"
hoelzl@40859: proof -
hoelzl@47694:   have X: "random_variable S X"
hoelzl@47694:     using Px by (auto simp: distributed_def)
hoelzl@47694:   have Y: "random_variable T Y"
hoelzl@47694:     using Py by (auto simp: distributed_def)
hoelzl@47694:   interpret S: sigma_finite_measure S by fact
hoelzl@47694:   interpret T: sigma_finite_measure T by fact
hoelzl@47694:   interpret ST: pair_sigma_finite S T ..
hoelzl@47694:   interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
hoelzl@47694:   interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
hoelzl@47694:   interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
wenzelm@53015:   let ?P = "S \<Otimes>\<^sub>M T"
hoelzl@47694:   let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"
hoelzl@47694: 
hoelzl@47694:   { fix A assume "A \<in> sets S"
hoelzl@47694:     with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
hoelzl@47694:       by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@47694:                intro!: arg_cong[where f="emeasure M"]) }
hoelzl@47694:   note marginal_eq1 = this
hoelzl@47694:   { fix A assume "A \<in> sets T"
hoelzl@47694:     with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
hoelzl@47694:       by (auto simp: emeasure_distr measurable_Pair measurable_space
hoelzl@47694:                intro!: arg_cong[where f="emeasure M"]) }
hoelzl@47694:   note marginal_eq2 = this
hoelzl@47694: 
hoelzl@47694:   have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
hoelzl@47694:     by auto
hoelzl@47694: 
wenzelm@53015:   have distr_eq: "distr M S X \<Otimes>\<^sub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
hoelzl@47694:     unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
hoelzl@47694:   proof (subst pair_measure_density)
hoelzl@47694:     show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
hoelzl@47694:       "AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
hoelzl@47694:       using Px Py by (auto simp: distributed_def)
hoelzl@47694:     show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
hoelzl@47694:   qed (fact | simp)+
hoelzl@47694:   
hoelzl@47694:   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:     unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..
hoelzl@47694: 
hoelzl@47694:   from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
hoelzl@47694:     by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
hoelzl@47694:   have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
hoelzl@47694:   proof (rule ST.AE_pair_measure)
hoelzl@47694:     show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
hoelzl@47694:       using f by auto
hoelzl@47694:     show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
hoelzl@47694:       using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
hoelzl@47694:   qed
hoelzl@47694: 
hoelzl@47694:   have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@47694:     by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@47694:   moreover
hoelzl@47694:   have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
hoelzl@47694:     by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@47694:   ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by eventually_elim auto
hoelzl@47694: 
hoelzl@47694:   show "?M = ?R"
hoelzl@47694:     unfolding M f_def
hoelzl@47694:     using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
hoelzl@47694:     by (rule ST.KL_density_density)
hoelzl@47694: 
wenzelm@53015:   assume int: "integrable (S \<Otimes>\<^sub>M T) f"
hoelzl@47694:   show "0 \<le> ?M" unfolding M
hoelzl@47694:   proof (rule ST.KL_density_density_nonneg
hoelzl@47694:     [OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
wenzelm@53015:     show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x))) "
hoelzl@47694:       unfolding distributed_distr_eq_density[OF Pxy, symmetric]
hoelzl@47694:       using distributed_measurable[OF Pxy] by (rule prob_space_distr)
wenzelm@53015:     show "prob_space (density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
hoelzl@47694:       unfolding distr_eq[symmetric] by unfold_locales
hoelzl@40859:   qed
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma (in information_space)
hoelzl@47694:   fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694:   assumes "sigma_finite_measure S" "sigma_finite_measure T"
hoelzl@47694:   assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:   assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
hoelzl@47694:   shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
hoelzl@36624: proof -
hoelzl@47694:   interpret S: sigma_finite_measure S by fact
hoelzl@47694:   interpret T: sigma_finite_measure T by fact
hoelzl@47694:   interpret ST: pair_sigma_finite S T ..
hoelzl@36080: 
wenzelm@53015:   have "AE x in S \<Otimes>\<^sub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
hoelzl@47694:   moreover
wenzelm@53015:   have "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
hoelzl@47694:   moreover 
wenzelm@53015:   have "AE x in S \<Otimes>\<^sub>M T. Pxy x = Px (fst x) * Py (snd x)"
hoelzl@47694:     using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy]
hoelzl@47694:     by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'')
wenzelm@53015:   ultimately have "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
hoelzl@47694:     by eventually_elim simp
wenzelm@53015:   then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^sub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@47694:     by (rule integral_cong_AE)
hoelzl@47694:   then show ?thesis
hoelzl@47694:     by (subst mutual_information_distr[OF assms(1-5)]) simp
hoelzl@36080: qed
hoelzl@36080: 
hoelzl@47694: lemma (in information_space) mutual_information_simple_distributed:
hoelzl@47694:   assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py"
hoelzl@47694:   assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:   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: proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@47694:   note fin = simple_distributed_joint_finite[OF XY, simp]
hoelzl@47694:   show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
hoelzl@47694:   let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@47694:   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:     by auto
wenzelm@53015:   with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M))) =
hoelzl@47694:     (\<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:     by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta'
hoelzl@47694:              intro!: setsum_cong)
hoelzl@47694: qed
hoelzl@36080: 
hoelzl@47694: lemma (in information_space)
hoelzl@47694:   fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694:   assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py"
hoelzl@47694:   assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:   assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
hoelzl@47694:   shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0"
hoelzl@47694: proof (subst mutual_information_simple_distributed[OF Px Py Pxy])
hoelzl@47694:   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:     (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)"
hoelzl@47694:     by (intro setsum_cong) (auto simp: ae)
hoelzl@47694:   then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M.
hoelzl@47694:     Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
hoelzl@47694: qed
hoelzl@36080: 
hoelzl@39097: subsection {* Entropy *}
hoelzl@39097: 
hoelzl@47694: definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where
hoelzl@47694:   "entropy b S X = - KL_divergence b S (distr M S X)"
hoelzl@47694: 
hoelzl@40859: abbreviation (in information_space)
hoelzl@40859:   entropy_Pow ("\<H>'(_')") where
hoelzl@47694:   "\<H>(X) \<equiv> entropy b (count_space (X`space M)) X"
hoelzl@41981: 
hoelzl@49791: lemma (in prob_space) distributed_RN_deriv:
hoelzl@49791:   assumes X: "distributed M S X Px"
hoelzl@49791:   shows "AE x in S. RN_deriv S (density S Px) x = Px x"
hoelzl@49791: proof -
hoelzl@49791:   note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49791:   interpret X: prob_space "distr M S X"
hoelzl@49791:     using D(1) by (rule prob_space_distr)
hoelzl@49791: 
hoelzl@49791:   have sf: "sigma_finite_measure (distr M S X)" by default
hoelzl@49791:   show ?thesis
hoelzl@49791:     using D
hoelzl@49791:     apply (subst eq_commute)
hoelzl@49791:     apply (intro RN_deriv_unique_sigma_finite)
hoelzl@49791:     apply (auto intro: divide_nonneg_nonneg measure_nonneg
hoelzl@49791:              simp: distributed_distr_eq_density[symmetric, OF X] sf)
hoelzl@49791:     done
hoelzl@49791: qed
hoelzl@49791: 
hoelzl@49788: lemma (in information_space)
hoelzl@47694:   fixes X :: "'a \<Rightarrow> 'b"
hoelzl@49785:   assumes X: "distributed M MX X f"
hoelzl@49788:   shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)
hoelzl@49788: proof -
hoelzl@49785:   note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49791:   note ae = distributed_RN_deriv[OF X]
hoelzl@49788: 
hoelzl@49788:   have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) =
hoelzl@49785:     log b (f x)"
hoelzl@49785:     unfolding distributed_distr_eq_density[OF X]
hoelzl@49785:     apply (subst AE_density)
hoelzl@49785:     using D apply simp
hoelzl@49785:     using ae apply eventually_elim
hoelzl@49785:     apply auto
hoelzl@49785:     done
hoelzl@49788: 
hoelzl@49788:   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:     unfolding distributed_distr_eq_density[OF X]
hoelzl@49785:     using D
hoelzl@49785:     by (subst integral_density)
hoelzl@49785:        (auto simp: borel_measurable_ereal_iff)
hoelzl@49788: 
hoelzl@49788:   show ?eq
hoelzl@49788:     unfolding entropy_def KL_divergence_def entropy_density_def comp_def
hoelzl@49788:     apply (subst integral_cong_AE)
hoelzl@49788:     apply (rule ae_eq)
hoelzl@49788:     apply (rule int_eq)
hoelzl@49788:     done
hoelzl@49788: qed
hoelzl@49785: 
hoelzl@49786: lemma (in prob_space) distributed_imp_emeasure_nonzero:
hoelzl@49786:   assumes X: "distributed M MX X Px"
hoelzl@49786:   shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0"
hoelzl@49786: proof
hoelzl@49786:   note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786:   interpret X: prob_space "distr M MX X"
hoelzl@49786:     using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@49786: 
hoelzl@49786:   assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0"
hoelzl@49786:   with Px have "AE x in MX. Px x = 0"
hoelzl@49786:     by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ereal_iff)
hoelzl@49786:   moreover
wenzelm@53015:   from X.emeasure_space_1 have "(\<integral>\<^sup>+x. Px x \<partial>MX) = 1"
hoelzl@49786:     unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786:     by (subst (asm) emeasure_density)
hoelzl@49786:        (auto simp: borel_measurable_ereal_iff intro!: integral_cong cong: positive_integral_cong)
hoelzl@49786:   ultimately show False
hoelzl@49786:     by (simp add: positive_integral_cong_AE)
hoelzl@49786: qed
hoelzl@49786: 
hoelzl@49786: lemma (in information_space) entropy_le:
hoelzl@49786:   fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
hoelzl@49786:   assumes X: "distributed M MX X Px"
hoelzl@49786:   and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>"
hoelzl@49786:   and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786:   shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
hoelzl@49786: proof -
hoelzl@49786:   note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786:   interpret X: prob_space "distr M MX X"
hoelzl@49786:     using distributed_measurable[OF X] by (rule prob_space_distr)
hoelzl@49786: 
hoelzl@49786:   have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) = 
hoelzl@49786:     - log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)"
hoelzl@49786:     using Px fin
hoelzl@49786:     by (subst integral_indicator) (auto simp: measure_def borel_measurable_ereal_iff)
hoelzl@49786:   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:     unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786:     apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus])
hoelzl@49786:     by (subst integral_density) (auto simp: borel_measurable_ereal_iff intro!: integral_cong)
hoelzl@49786:   also have "\<dots> \<le> (\<integral> x. - log b (1 / Px x) \<partial>distr M MX X)"
hoelzl@49786:   proof (rule X.jensens_inequality[of "\<lambda>x. 1 / Px x" "{0<..}" 0 1 "\<lambda>x. - log b x"])
hoelzl@49786:     show "AE x in distr M MX X. 1 / Px x \<in> {0<..}"
hoelzl@49786:       unfolding distributed_distr_eq_density[OF X]
hoelzl@49786:       using Px by (auto simp: AE_density)
hoelzl@49786:     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:       by (auto simp: one_ereal_def)
wenzelm@53015:     have "(\<integral>\<^sup>+ x. max 0 (ereal (- (if Px x = 0 then 0 else 1))) \<partial>MX) = (\<integral>\<^sup>+ x. 0 \<partial>MX)"
hoelzl@49786:       by (intro positive_integral_cong) (auto split: split_max)
hoelzl@49786:     then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)"
hoelzl@49786:       unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786:       by (auto simp: positive_integral_density integrable_def borel_measurable_ereal_iff fin positive_integral_max_0
hoelzl@49786:               cong: positive_integral_cong)
hoelzl@49786:     have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) =
hoelzl@49786:       integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786:       using Px
hoelzl@49786:       by (intro integrable_cong_AE)
hoelzl@49786:          (auto simp: borel_measurable_ereal_iff log_divide_eq
hoelzl@49786:                   intro!: measurable_If)
hoelzl@49786:     then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))"
hoelzl@49786:       unfolding distributed_distr_eq_density[OF X]
hoelzl@49786:       using Px int
hoelzl@49786:       by (subst integral_density) (auto simp: borel_measurable_ereal_iff)
hoelzl@49786:   qed (auto simp: minus_log_convex[OF b_gt_1])
hoelzl@49786:   also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)"
hoelzl@49786:     unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786:     by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq)
hoelzl@49786:   also have "\<dots> = - entropy b MX X"
hoelzl@49786:     unfolding distributed_distr_eq_density[OF X] using Px
hoelzl@49786:     by (subst entropy_distr[OF X]) (auto simp: borel_measurable_ereal_iff integral_density)
hoelzl@49786:   finally show ?thesis
hoelzl@49786:     by simp
hoelzl@49786: qed
hoelzl@49786: 
hoelzl@49786: lemma (in information_space) entropy_le_space:
hoelzl@49786:   fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
hoelzl@49786:   assumes X: "distributed M MX X Px"
hoelzl@49786:   and fin: "finite_measure MX"
hoelzl@49786:   and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
hoelzl@49786:   shows "entropy b MX X \<le> log b (measure MX (space MX))"
hoelzl@49786: proof -
hoelzl@49786:   note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
hoelzl@49786:   interpret finite_measure MX by fact
hoelzl@49786:   have "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
hoelzl@49786:     using int X by (intro entropy_le) auto
hoelzl@49786:   also have "\<dots> \<le> log b (measure MX (space MX))"
hoelzl@49786:     using Px distributed_imp_emeasure_nonzero[OF X]
hoelzl@49786:     by (intro log_le)
hoelzl@49786:        (auto intro!: borel_measurable_ereal_iff finite_measure_mono b_gt_1
hoelzl@49786:                      less_le[THEN iffD2] measure_nonneg simp: emeasure_eq_measure)
hoelzl@49786:   finally show ?thesis .
hoelzl@49786: qed
hoelzl@49786: 
hoelzl@47694: lemma (in information_space) entropy_uniform:
hoelzl@49785:   assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f")
hoelzl@47694:   shows "entropy b MX X = log b (measure MX A)"
hoelzl@49785: proof (subst entropy_distr[OF X])
hoelzl@49785:   have [simp]: "emeasure MX A \<noteq> \<infinity>"
hoelzl@49785:     using uniform_distributed_params[OF X] by (auto simp add: measure_def)
hoelzl@49785:   have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
hoelzl@49785:     (\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)"
hoelzl@49785:     using measure_nonneg[of MX A] uniform_distributed_params[OF X]
hoelzl@49785:     by (auto intro!: integral_cong split: split_indicator simp: log_divide_eq)
hoelzl@49785:   show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
hoelzl@49785:     log b (measure MX A)"
hoelzl@49785:     unfolding eq using uniform_distributed_params[OF X]
hoelzl@49785:     by (subst lebesgue_integral_cmult) (auto simp: measure_def)
hoelzl@49785: qed
hoelzl@36080: 
hoelzl@47694: lemma (in information_space) entropy_simple_distributed:
hoelzl@49786:   "simple_distributed M X f \<Longrightarrow> \<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))"
hoelzl@49786:   by (subst entropy_distr[OF simple_distributed])
hoelzl@49786:      (auto simp add: lebesgue_integral_count_space_finite)
hoelzl@39097: 
hoelzl@40859: lemma (in information_space) entropy_le_card_not_0:
hoelzl@47694:   assumes X: "simple_distributed M X f"
hoelzl@47694:   shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))"
hoelzl@39097: proof -
hoelzl@49787:   let ?X = "count_space (X`space M)"
hoelzl@49787:   have "\<H>(X) \<le> log b (measure ?X {x \<in> space ?X. f x \<noteq> 0})"
hoelzl@49787:     by (rule entropy_le[OF simple_distributed[OF X]])
hoelzl@49787:        (simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space)
hoelzl@49787:   also have "measure ?X {x \<in> space ?X. f x \<noteq> 0} = card (X ` space M \<inter> {x. f x \<noteq> 0})"
hoelzl@49787:     by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def Int_def)
hoelzl@49787:   finally show ?thesis .
hoelzl@39097: qed
hoelzl@39097: 
hoelzl@40859: lemma (in information_space) entropy_le_card:
hoelzl@49787:   assumes X: "simple_distributed M X f"
hoelzl@40859:   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
hoelzl@49787: proof -
hoelzl@49787:   let ?X = "count_space (X`space M)"
hoelzl@49787:   have "\<H>(X) \<le> log b (measure ?X (space ?X))"
hoelzl@49787:     by (rule entropy_le_space[OF simple_distributed[OF X]])
hoelzl@49787:        (simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space finite_measure_count_space)
hoelzl@49787:   also have "measure ?X (space ?X) = card (X ` space M)"
hoelzl@49787:     by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def)
hoelzl@39097:   finally show ?thesis .
hoelzl@39097: qed
hoelzl@39097: 
hoelzl@39097: subsection {* Conditional Mutual Information *}
hoelzl@39097: 
hoelzl@36080: definition (in prob_space)
hoelzl@41689:   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
wenzelm@53015:     mutual_information b MX (MY \<Otimes>\<^sub>M MZ) X (\<lambda>x. (Y x, Z x)) -
hoelzl@41689:     mutual_information b MX MZ X Z"
hoelzl@36080: 
hoelzl@40859: abbreviation (in information_space)
hoelzl@40859:   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
hoelzl@36624:   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
hoelzl@47694:     (count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z"
hoelzl@36080: 
hoelzl@49787: lemma (in information_space)
hoelzl@47694:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
hoelzl@50003:   assumes Px[measurable]: "distributed M S X Px"
hoelzl@50003:   assumes Pz[measurable]: "distributed M P Z Pz"
wenzelm@53015:   assumes Pyz[measurable]: "distributed M (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
wenzelm@53015:   assumes Pxz[measurable]: "distributed M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) Pxz"
wenzelm@53015:   assumes Pxyz[measurable]: "distributed M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
wenzelm@53015:   assumes I1: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
wenzelm@53015:   assumes I2: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
hoelzl@49787:   shows conditional_mutual_information_generic_eq: "conditional_mutual_information b S T P X Y Z
wenzelm@53015:     = (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" (is "?eq")
hoelzl@49787:     and conditional_mutual_information_generic_nonneg: "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
hoelzl@40859: proof -
hoelzl@47694:   interpret S: sigma_finite_measure S by fact
hoelzl@47694:   interpret T: sigma_finite_measure T by fact
hoelzl@47694:   interpret P: sigma_finite_measure P by fact
hoelzl@47694:   interpret TP: pair_sigma_finite T P ..
hoelzl@47694:   interpret SP: pair_sigma_finite S P ..
hoelzl@49787:   interpret ST: pair_sigma_finite S T ..
wenzelm@53015:   interpret SPT: pair_sigma_finite "S \<Otimes>\<^sub>M P" T ..
wenzelm@53015:   interpret STP: pair_sigma_finite S "T \<Otimes>\<^sub>M P" ..
wenzelm@53015:   interpret TPS: pair_sigma_finite "T \<Otimes>\<^sub>M P" S ..
wenzelm@53015:   have TP: "sigma_finite_measure (T \<Otimes>\<^sub>M P)" ..
wenzelm@53015:   have SP: "sigma_finite_measure (S \<Otimes>\<^sub>M P)" ..
wenzelm@53015:   have YZ: "random_variable (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x))"
hoelzl@47694:     using Pyz by (simp add: distributed_measurable)
hoelzl@47694:   
wenzelm@53015:   from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) =
wenzelm@53015:     distr (distr M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). (x, z))"
hoelzl@50003:     by (simp add: comp_def distr_distr)
hoelzl@40859: 
hoelzl@47694:   have "mutual_information b S P X Z =
wenzelm@53015:     (\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^sub>M P))"
hoelzl@47694:     by (rule mutual_information_distr[OF S P Px Pz Pxz])
wenzelm@53015:   also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))"
hoelzl@47694:     using b_gt_1 Pxz Px Pz
hoelzl@50003:     by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) (auto simp: split_beta')
hoelzl@47694:   finally have mi_eq:
wenzelm@53015:     "mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" .
hoelzl@47694:   
wenzelm@53015:   have ae1: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@47694:     by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
wenzelm@53015:   moreover have ae2: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
wenzelm@53015:   moreover have ae3: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
wenzelm@53015:   moreover have ae4: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
wenzelm@53015:   moreover have ae5: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Px (fst x)"
hoelzl@50003:     using Px by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
wenzelm@53015:   moreover have ae6: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pyz (snd x)"
hoelzl@50003:     using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
wenzelm@53015:   moreover have ae7: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd (snd x))"
hoelzl@50003:     using Pz Pz[THEN distributed_real_measurable]
hoelzl@50003:     by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
wenzelm@53015:   moreover have ae8: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
hoelzl@47694:     using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
hoelzl@50003:     by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure)
hoelzl@47694:   moreover note Pxyz[THEN distributed_real_AE]
wenzelm@53015:   ultimately have ae: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P.
hoelzl@47694:     Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
hoelzl@47694:     Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
hoelzl@47694:     Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
hoelzl@47694:   proof eventually_elim
hoelzl@47694:     case (goal1 x)
hoelzl@47694:     show ?case
hoelzl@40859:     proof cases
hoelzl@47694:       assume "Pxyz x \<noteq> 0"
hoelzl@47694:       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:         by auto
hoelzl@47694:       then show ?thesis
hoelzl@47694:         using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@40859:     qed simp
hoelzl@40859:   qed
hoelzl@49787:   with I1 I2 show ?eq
hoelzl@40859:     unfolding conditional_mutual_information_def
hoelzl@47694:     apply (subst mi_eq)
hoelzl@47694:     apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
hoelzl@47694:     apply (subst integral_diff(2)[symmetric])
hoelzl@47694:     apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
hoelzl@47694:     done
hoelzl@49787: 
wenzelm@53015:   let ?P = "density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) Pxyz"
hoelzl@49787:   interpret P: prob_space ?P
hoelzl@49787:     unfolding distributed_distr_eq_density[OF Pxyz, symmetric]
hoelzl@50003:     by (rule prob_space_distr) simp
hoelzl@49787: 
wenzelm@53015:   let ?Q = "density (T \<Otimes>\<^sub>M P) Pyz"
hoelzl@49787:   interpret Q: prob_space ?Q
hoelzl@49787:     unfolding distributed_distr_eq_density[OF Pyz, symmetric]
hoelzl@50003:     by (rule prob_space_distr) simp
hoelzl@49787: 
hoelzl@49787:   let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"
hoelzl@49787: 
hoelzl@49787:   from subdensity_real[of snd, OF _ Pyz Pz]
wenzelm@53015:   have aeX1: "AE x in T \<Otimes>\<^sub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
wenzelm@53015:   have aeX2: "AE x in T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd x)"
hoelzl@50003:     using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
hoelzl@49787: 
wenzelm@53015:   have aeX3: "AE y in T \<Otimes>\<^sub>M P. (\<integral>\<^sup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
hoelzl@49788:     using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
hoelzl@50003:     by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49787: 
wenzelm@53015:   have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^sup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))"
hoelzl@49787:     apply (subst positive_integral_density)
hoelzl@50003:     apply simp
hoelzl@49787:     apply (rule distributed_AE[OF Pxyz])
hoelzl@50003:     apply auto []
hoelzl@49787:     apply (rule positive_integral_mono_AE)
hoelzl@49787:     using ae5 ae6 ae7 ae8
hoelzl@49787:     apply eventually_elim
hoelzl@49787:     apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
hoelzl@49787:     done
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)"
hoelzl@50003:     by (subst STP.positive_integral_snd_measurable[symmetric]) (auto simp add: split_beta')
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)"
hoelzl@49787:     apply (rule positive_integral_cong_AE)
hoelzl@49787:     using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
hoelzl@49787:     apply eventually_elim
hoelzl@49787:   proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
hoelzl@49787:     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"
wenzelm@53015:       "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" 
wenzelm@53015:     then show "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
hoelzl@50003:       by (subst positive_integral_multc)
hoelzl@50003:          (auto intro!: divide_nonneg_nonneg split: prod.split)
hoelzl@49787:   qed
hoelzl@49787:   also have "\<dots> = 1"
hoelzl@49787:     using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
hoelzl@50003:     by (subst positive_integral_density[symmetric]) auto
wenzelm@53015:   finally have le1: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> 1" .
hoelzl@49787:   also have "\<dots> < \<infinity>" by simp
wenzelm@53015:   finally have fin: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp
hoelzl@49787: 
wenzelm@53015:   have pos: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> 0"
hoelzl@49787:     apply (subst positive_integral_density)
hoelzl@50003:     apply simp
hoelzl@49787:     apply (rule distributed_AE[OF Pxyz])
hoelzl@50003:     apply auto []
hoelzl@49787:     apply (simp add: split_beta')
hoelzl@49787:   proof
hoelzl@49787:     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)))"
wenzelm@53015:     assume "(\<integral>\<^sup>+ x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0"
wenzelm@53015:     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. ?g x \<le> 0"
hoelzl@50003:       by (intro positive_integral_0_iff_AE[THEN iffD1]) auto
wenzelm@53015:     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x = 0"
hoelzl@49787:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787:       by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
wenzelm@53015:     then have "(\<integral>\<^sup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) = 0"
hoelzl@49787:       by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
hoelzl@49787:     with P.emeasure_space_1 show False
hoelzl@50003:       by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
hoelzl@49787:   qed
hoelzl@49787: 
wenzelm@53015:   have neg: "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) = 0"
hoelzl@49787:     apply (rule positive_integral_0_iff_AE[THEN iffD2])
hoelzl@50003:     apply simp
hoelzl@49787:     apply (subst AE_density)
hoelzl@50003:     apply simp
hoelzl@49787:     using ae5 ae6 ae7 ae8
hoelzl@49787:     apply eventually_elim
hoelzl@49787:     apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
hoelzl@49787:     done
hoelzl@49787: 
wenzelm@53015:   have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
hoelzl@49787:     apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
hoelzl@49787:     using ae
hoelzl@50003:     apply (auto simp: split_beta')
hoelzl@49787:     done
hoelzl@49787: 
wenzelm@53015:   have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)"
hoelzl@49787:   proof (intro le_imp_neg_le log_le[OF b_gt_1])
wenzelm@53015:     show "0 < integral\<^sup>L ?P ?f"
hoelzl@49787:       using neg pos fin positive_integral_positive[of ?P ?f]
wenzelm@53015:       by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
wenzelm@53015:     show "integral\<^sup>L ?P ?f \<le> 1"
hoelzl@49787:       using neg le1 fin positive_integral_positive[of ?P ?f]
wenzelm@53015:       by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
hoelzl@49787:   qed
wenzelm@53015:   also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
hoelzl@49787:   proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
hoelzl@49787:     show "AE x in ?P. ?f x \<in> {0<..}"
hoelzl@49787:       unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
hoelzl@49787:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787:       by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
hoelzl@49787:     show "integrable ?P ?f"
hoelzl@49787:       unfolding integrable_def 
hoelzl@50003:       using fin neg by (auto simp: split_beta')
hoelzl@49787:     show "integrable ?P (\<lambda>x. - log b (?f x))"
hoelzl@49787:       apply (subst integral_density)
hoelzl@50003:       apply simp
hoelzl@50003:       apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003:       apply simp
hoelzl@49787:       apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
hoelzl@50003:       apply simp
hoelzl@50003:       apply simp
hoelzl@49787:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787:       apply eventually_elim
hoelzl@49787:       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:       done
hoelzl@49787:   qed (auto simp: b_gt_1 minus_log_convex)
hoelzl@49787:   also have "\<dots> = conditional_mutual_information b S T P X Y Z"
hoelzl@49787:     unfolding `?eq`
hoelzl@49787:     apply (subst integral_density)
hoelzl@50003:     apply simp
hoelzl@50003:     apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003:     apply simp
hoelzl@49787:     apply (intro integral_cong_AE)
hoelzl@49787:     using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49787:     apply eventually_elim
hoelzl@49787:     apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49787:     done
hoelzl@49787:   finally show ?nonneg
hoelzl@49787:     by simp
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@49803: lemma (in information_space)
hoelzl@49803:   fixes Px :: "_ \<Rightarrow> real"
hoelzl@49803:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
hoelzl@49803:   assumes Fx: "finite_entropy S X Px"
hoelzl@49803:   assumes Fz: "finite_entropy P Z Pz"
wenzelm@53015:   assumes Fyz: "finite_entropy (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
wenzelm@53015:   assumes Fxz: "finite_entropy (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) Pxz"
wenzelm@53015:   assumes Fxyz: "finite_entropy (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
hoelzl@49803:   shows conditional_mutual_information_generic_eq': "conditional_mutual_information b S T P X Y Z
wenzelm@53015:     = (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" (is "?eq")
hoelzl@49803:     and conditional_mutual_information_generic_nonneg': "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
hoelzl@49803: proof -
hoelzl@50003:   note Px = Fx[THEN finite_entropy_distributed, measurable]
hoelzl@50003:   note Pz = Fz[THEN finite_entropy_distributed, measurable]
hoelzl@50003:   note Pyz = Fyz[THEN finite_entropy_distributed, measurable]
hoelzl@50003:   note Pxz = Fxz[THEN finite_entropy_distributed, measurable]
hoelzl@50003:   note Pxyz = Fxyz[THEN finite_entropy_distributed, measurable]
hoelzl@49803: 
hoelzl@49803:   interpret S: sigma_finite_measure S by fact
hoelzl@49803:   interpret T: sigma_finite_measure T by fact
hoelzl@49803:   interpret P: sigma_finite_measure P by fact
hoelzl@49803:   interpret TP: pair_sigma_finite T P ..
hoelzl@49803:   interpret SP: pair_sigma_finite S P ..
hoelzl@49803:   interpret ST: pair_sigma_finite S T ..
wenzelm@53015:   interpret SPT: pair_sigma_finite "S \<Otimes>\<^sub>M P" T ..
wenzelm@53015:   interpret STP: pair_sigma_finite S "T \<Otimes>\<^sub>M P" ..
wenzelm@53015:   interpret TPS: pair_sigma_finite "T \<Otimes>\<^sub>M P" S ..
wenzelm@53015:   have TP: "sigma_finite_measure (T \<Otimes>\<^sub>M P)" ..
wenzelm@53015:   have SP: "sigma_finite_measure (S \<Otimes>\<^sub>M P)" ..
hoelzl@49803: 
wenzelm@53015:   from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) =
wenzelm@53015:     distr (distr M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). (x, z))"
hoelzl@50003:     by (simp add: distr_distr comp_def)
hoelzl@49803: 
hoelzl@49803:   have "mutual_information b S P X Z =
wenzelm@53015:     (\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^sub>M P))"
hoelzl@49803:     by (rule mutual_information_distr[OF S P Px Pz Pxz])
wenzelm@53015:   also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))"
hoelzl@49803:     using b_gt_1 Pxz Px Pz
hoelzl@49803:     by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"])
hoelzl@50003:        (auto simp: split_beta')
hoelzl@49803:   finally have mi_eq:
wenzelm@53015:     "mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))" .
hoelzl@49803:   
wenzelm@53015:   have ae1: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@49803:     by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
wenzelm@53015:   moreover have ae2: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
wenzelm@53015:   moreover have ae3: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
wenzelm@53015:   moreover have ae4: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
hoelzl@50003:     by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
wenzelm@53015:   moreover have ae5: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Px (fst x)"
hoelzl@50003:     using Px by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
wenzelm@53015:   moreover have ae6: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pyz (snd x)"
hoelzl@50003:     using Pyz by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
wenzelm@53015:   moreover have ae7: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd (snd x))"
hoelzl@50003:     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)
wenzelm@53015:   moreover have ae8: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
hoelzl@49803:     using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
hoelzl@49803:     by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def)
hoelzl@49803:   moreover note ae9 = Pxyz[THEN distributed_real_AE]
wenzelm@53015:   ultimately have ae: "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P.
hoelzl@49803:     Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
hoelzl@49803:     Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
hoelzl@49803:     Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
hoelzl@49803:   proof eventually_elim
hoelzl@49803:     case (goal1 x)
hoelzl@49803:     show ?case
hoelzl@49803:     proof cases
hoelzl@49803:       assume "Pxyz x \<noteq> 0"
hoelzl@49803:       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:         by auto
hoelzl@49803:       then show ?thesis
hoelzl@49803:         using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@49803:     qed simp
hoelzl@49803:   qed
hoelzl@49803: 
wenzelm@53015:   have "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)
hoelzl@49803:     (\<lambda>x. Pxyz x * log b (Pxyz x) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pyz (snd x)))"
hoelzl@49803:     using finite_entropy_integrable[OF Fxyz]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fyz Pxyz, of snd]
hoelzl@49803:     by simp
wenzelm@53015:   moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z)))) \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)"
hoelzl@50003:     using Pxyz Px Pyz by simp
wenzelm@53015:   ultimately have I1: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
hoelzl@49803:     apply (rule integrable_cong_AE_imp)
hoelzl@49803:     using ae1 ae4 ae5 ae6 ae9
hoelzl@49803:     by eventually_elim
hoelzl@49803:        (auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)
hoelzl@49803: 
wenzelm@53015:   have "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)
hoelzl@49803:     (\<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:     using finite_entropy_integrable_transform[OF Fxz Pxyz, of "\<lambda>x. (fst x, snd (snd x))"]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
hoelzl@49803:     using finite_entropy_integrable_transform[OF Fz Pxyz, of "snd \<circ> snd"]
hoelzl@50003:     by simp
wenzelm@53015:   moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z))) \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)"
hoelzl@49803:     using Pxyz Px Pz
hoelzl@50003:     by auto
wenzelm@53015:   ultimately have I2: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
hoelzl@49803:     apply (rule integrable_cong_AE_imp)
hoelzl@49803:     using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 ae9
hoelzl@49803:     by eventually_elim
hoelzl@49803:        (auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)
hoelzl@49803: 
hoelzl@49803:   from ae I1 I2 show ?eq
hoelzl@49803:     unfolding conditional_mutual_information_def
hoelzl@49803:     apply (subst mi_eq)
hoelzl@49803:     apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
hoelzl@49803:     apply (subst integral_diff(2)[symmetric])
hoelzl@49803:     apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
hoelzl@49803:     done
hoelzl@49803: 
wenzelm@53015:   let ?P = "density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) Pxyz"
hoelzl@49803:   interpret P: prob_space ?P
hoelzl@50003:     unfolding distributed_distr_eq_density[OF Pxyz, symmetric] by (rule prob_space_distr) simp
hoelzl@49803: 
wenzelm@53015:   let ?Q = "density (T \<Otimes>\<^sub>M P) Pyz"
hoelzl@49803:   interpret Q: prob_space ?Q
hoelzl@50003:     unfolding distributed_distr_eq_density[OF Pyz, symmetric] by (rule prob_space_distr) simp
hoelzl@49803: 
hoelzl@49803:   let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"
hoelzl@49803: 
hoelzl@49803:   from subdensity_real[of snd, OF _ Pyz Pz]
wenzelm@53015:   have aeX1: "AE x in T \<Otimes>\<^sub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
wenzelm@53015:   have aeX2: "AE x in T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd x)"
hoelzl@50003:     using Pz by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
hoelzl@49803: 
wenzelm@53015:   have aeX3: "AE y in T \<Otimes>\<^sub>M P. (\<integral>\<^sup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
hoelzl@49803:     using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
hoelzl@50003:     by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
wenzelm@53015:   have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^sup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P))"
hoelzl@49803:     apply (subst positive_integral_density)
hoelzl@49803:     apply (rule distributed_borel_measurable[OF Pxyz])
hoelzl@49803:     apply (rule distributed_AE[OF Pxyz])
hoelzl@50003:     apply simp
hoelzl@49803:     apply (rule positive_integral_mono_AE)
hoelzl@49803:     using ae5 ae6 ae7 ae8
hoelzl@49803:     apply eventually_elim
hoelzl@49803:     apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
hoelzl@49803:     done
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)"
hoelzl@49803:     by (subst STP.positive_integral_snd_measurable[symmetric])
hoelzl@50003:        (auto simp add: split_beta')
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)"
hoelzl@49803:     apply (rule positive_integral_cong_AE)
hoelzl@49803:     using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
hoelzl@49803:     apply eventually_elim
hoelzl@49803:   proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
hoelzl@49803:     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"
wenzelm@53015:       "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)" 
wenzelm@53015:     then show "(\<integral>\<^sup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
hoelzl@50003:       by (subst positive_integral_multc) (auto intro!: divide_nonneg_nonneg)
hoelzl@49803:   qed
hoelzl@49803:   also have "\<dots> = 1"
hoelzl@49803:     using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
hoelzl@50003:     by (subst positive_integral_density[symmetric]) auto
wenzelm@53015:   finally have le1: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<le> 1" .
hoelzl@49803:   also have "\<dots> < \<infinity>" by simp
wenzelm@53015:   finally have fin: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp
hoelzl@49803: 
wenzelm@53015:   have pos: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> 0"
hoelzl@49803:     apply (subst positive_integral_density)
hoelzl@50003:     apply simp
hoelzl@49803:     apply (rule distributed_AE[OF Pxyz])
hoelzl@50003:     apply simp
hoelzl@49803:     apply (simp add: split_beta')
hoelzl@49803:   proof
hoelzl@49803:     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)))"
wenzelm@53015:     assume "(\<integral>\<^sup>+ x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0"
wenzelm@53015:     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. ?g x \<le> 0"
hoelzl@50003:       by (intro positive_integral_0_iff_AE[THEN iffD1]) (auto intro!: borel_measurable_ereal measurable_If)
wenzelm@53015:     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x = 0"
hoelzl@49803:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803:       by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
wenzelm@53015:     then have "(\<integral>\<^sup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) = 0"
hoelzl@49803:       by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
hoelzl@49803:     with P.emeasure_space_1 show False
hoelzl@50003:       by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
hoelzl@49803:   qed
hoelzl@49803: 
wenzelm@53015:   have neg: "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) = 0"
hoelzl@49803:     apply (rule positive_integral_0_iff_AE[THEN iffD2])
hoelzl@50003:     apply (auto simp: split_beta') []
hoelzl@49803:     apply (subst AE_density)
hoelzl@50003:     apply (auto simp: split_beta') []
hoelzl@49803:     using ae5 ae6 ae7 ae8
hoelzl@49803:     apply eventually_elim
hoelzl@49803:     apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
hoelzl@49803:     done
hoelzl@49803: 
wenzelm@53015:   have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
hoelzl@49803:     apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
hoelzl@49803:     using ae
hoelzl@50003:     apply (auto simp: split_beta')
hoelzl@49803:     done
hoelzl@49803: 
wenzelm@53015:   have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)"
hoelzl@49803:   proof (intro le_imp_neg_le log_le[OF b_gt_1])
wenzelm@53015:     show "0 < integral\<^sup>L ?P ?f"
hoelzl@49803:       using neg pos fin positive_integral_positive[of ?P ?f]
wenzelm@53015:       by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
wenzelm@53015:     show "integral\<^sup>L ?P ?f \<le> 1"
hoelzl@49803:       using neg le1 fin positive_integral_positive[of ?P ?f]
wenzelm@53015:       by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
hoelzl@49803:   qed
wenzelm@53015:   also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
hoelzl@49803:   proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
hoelzl@49803:     show "AE x in ?P. ?f x \<in> {0<..}"
hoelzl@49803:       unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
hoelzl@49803:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803:       by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
hoelzl@49803:     show "integrable ?P ?f"
hoelzl@49803:       unfolding integrable_def 
hoelzl@50003:       using fin neg by (auto simp: split_beta')
hoelzl@49803:     show "integrable ?P (\<lambda>x. - log b (?f x))"
hoelzl@49803:       apply (subst integral_density)
hoelzl@50003:       apply simp
hoelzl@50003:       apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003:       apply simp
hoelzl@49803:       apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
hoelzl@50003:       apply simp
hoelzl@50003:       apply simp
hoelzl@49803:       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803:       apply eventually_elim
hoelzl@49803:       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:       done
hoelzl@49803:   qed (auto simp: b_gt_1 minus_log_convex)
hoelzl@49803:   also have "\<dots> = conditional_mutual_information b S T P X Y Z"
hoelzl@49803:     unfolding `?eq`
hoelzl@49803:     apply (subst integral_density)
hoelzl@50003:     apply simp
hoelzl@50003:     apply (auto intro!: distributed_real_AE[OF Pxyz]) []
hoelzl@50003:     apply simp
hoelzl@49803:     apply (intro integral_cong_AE)
hoelzl@49803:     using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
hoelzl@49803:     apply eventually_elim
hoelzl@49803:     apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
hoelzl@49803:     done
hoelzl@49803:   finally show ?nonneg
hoelzl@49803:     by simp
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@40859: lemma (in information_space) conditional_mutual_information_eq:
hoelzl@47694:   assumes Pz: "simple_distributed M Z Pz"
hoelzl@47694:   assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
hoelzl@47694:   assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
hoelzl@47694:   assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
hoelzl@47694:   shows "\<I>(X ; Y | Z) =
hoelzl@47694:    (\<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: proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _
hoelzl@47694:     simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz]
hoelzl@47694:     simple_distributed_joint2[OF Pxyz]])
hoelzl@47694:   note simple_distributed_joint2_finite[OF Pxyz, simp]
hoelzl@47694:   show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   show "sigma_finite_measure (count_space (Z ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
wenzelm@53015:   have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M) =
hoelzl@47694:       count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@47694:     (is "?P = ?C")
hoelzl@47694:     by (simp add: pair_measure_count_space)
hoelzl@40859: 
hoelzl@47694:   let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)"
wenzelm@53015:   have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Z ` space M))"
hoelzl@47694:     using simple_distributed_joint[OF Pxz] by (rule distributed_measurable)
hoelzl@47694:   from measurable_comp[OF this measurable_fst]
hoelzl@47694:   have "random_variable (count_space (X ` space M)) X"
hoelzl@47694:     by (simp add: comp_def)
hoelzl@47694:   then have "simple_function M X"    
hoelzl@50002:     unfolding simple_function_def by (auto simp: measurable_count_space_eq2)
hoelzl@47694:   then have "simple_distributed M X ?Px"
hoelzl@47694:     by (rule simple_distributedI) auto
hoelzl@47694:   then show "distributed M (count_space (X ` space M)) X ?Px"
hoelzl@47694:     by (rule simple_distributed)
hoelzl@47694: 
hoelzl@47694:   let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)"
hoelzl@47694:   let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)"
hoelzl@47694:   let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)"
hoelzl@47694:   show
hoelzl@47694:       "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))"
hoelzl@47694:       "integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))"
hoelzl@47694:     by (auto intro!: integrable_count_space simp: pair_measure_count_space)
hoelzl@47694:   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:   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:   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:     by (auto intro!: ext)
hoelzl@47694:   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:     by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta')
hoelzl@36624: qed
hoelzl@36624: 
hoelzl@47694: lemma (in information_space) conditional_mutual_information_nonneg:
hoelzl@47694:   assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z"
hoelzl@47694:   shows "0 \<le> \<I>(X ; Y | Z)"
hoelzl@47694: proof -
wenzelm@53015:   have [simp]: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M) =
hoelzl@49787:       count_space (X`space M \<times> Y`space M \<times> Z`space M)"
hoelzl@49787:     by (simp add: pair_measure_count_space X Y Z simple_functionD)
hoelzl@49787:   note sf = sigma_finite_measure_count_space_finite[OF simple_functionD(1)]
hoelzl@49787:   note sd = simple_distributedI[OF _ refl]
hoelzl@49787:   note sp = simple_function_Pair
hoelzl@49787:   show ?thesis
hoelzl@49787:    apply (rule conditional_mutual_information_generic_nonneg[OF sf[OF X] sf[OF Y] sf[OF Z]])
hoelzl@49787:    apply (rule simple_distributed[OF sd[OF X]])
hoelzl@49787:    apply (rule simple_distributed[OF sd[OF Z]])
hoelzl@49787:    apply (rule simple_distributed_joint[OF sd[OF sp[OF Y Z]]])
hoelzl@49787:    apply (rule simple_distributed_joint[OF sd[OF sp[OF X Z]]])
hoelzl@49787:    apply (rule simple_distributed_joint2[OF sd[OF sp[OF X sp[OF Y Z]]]])
hoelzl@49787:    apply (auto intro!: integrable_count_space simp: X Y Z simple_functionD)
hoelzl@49787:    done
hoelzl@36080: qed
hoelzl@36080: 
hoelzl@39097: subsection {* Conditional Entropy *}
hoelzl@39097: 
hoelzl@36080: definition (in prob_space)
wenzelm@53015:   "conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / 
wenzelm@53015:     real (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
hoelzl@36080: 
hoelzl@40859: abbreviation (in information_space)
hoelzl@40859:   conditional_entropy_Pow ("\<H>'(_ | _')") where
hoelzl@47694:   "\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y"
hoelzl@36080: 
hoelzl@49791: lemma (in information_space) conditional_entropy_generic_eq:
hoelzl@49791:   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@49791:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@50003:   assumes Py[measurable]: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015:   shows "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@49791: proof -
hoelzl@49791:   interpret S: sigma_finite_measure S by fact
hoelzl@49791:   interpret T: sigma_finite_measure T by fact
hoelzl@49791:   interpret ST: pair_sigma_finite S T ..
hoelzl@49791: 
wenzelm@53015:   have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) x)"
hoelzl@49791:     unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
hoelzl@49791:     unfolding distributed_distr_eq_density[OF Pxy]
hoelzl@49791:     using distributed_RN_deriv[OF Pxy]
hoelzl@49791:     by auto
hoelzl@49791:   moreover
wenzelm@53015:   have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real (RN_deriv T (distr M T Y) (snd x))"
hoelzl@49791:     unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
hoelzl@49791:     unfolding distributed_distr_eq_density[OF Py]
hoelzl@49791:     apply (rule ST.AE_pair_measure)
immler@50244:     apply (auto intro!: sets.sets_Collect borel_measurable_eq measurable_compose[OF _ distributed_real_measurable[OF Py]]
hoelzl@49791:                         distributed_real_measurable[OF Pxy] distributed_real_AE[OF Py]
hoelzl@49791:                         borel_measurable_real_of_ereal measurable_compose[OF _ borel_measurable_RN_deriv_density])
hoelzl@49791:     using distributed_RN_deriv[OF Py]
hoelzl@49791:     apply auto
hoelzl@49791:     done    
hoelzl@49791:   ultimately
wenzelm@53015:   have "conditional_entropy b S T X Y = - (\<integral>x. Pxy x * log b (Pxy x / Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@49791:     unfolding conditional_entropy_def neg_equal_iff_equal
hoelzl@49791:     apply (subst integral_density(1)[symmetric])
hoelzl@49791:     apply (auto simp: distributed_real_measurable[OF Pxy] distributed_real_AE[OF Pxy]
hoelzl@49791:                       measurable_compose[OF _ distributed_real_measurable[OF Py]]
hoelzl@49791:                       distributed_distr_eq_density[OF Pxy]
hoelzl@49791:                 intro!: integral_cong_AE)
hoelzl@49791:     done
hoelzl@49791:   then show ?thesis by (simp add: split_beta')
hoelzl@49791: qed
hoelzl@49791: 
hoelzl@49791: lemma (in information_space) conditional_entropy_eq_entropy:
hoelzl@47694:   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
hoelzl@47694:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@47694:   assumes Py: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015:   assumes I1: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
wenzelm@53015:   assumes I2: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
wenzelm@53015:   shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
hoelzl@40859: proof -
hoelzl@47694:   interpret S: sigma_finite_measure S by fact
hoelzl@47694:   interpret T: sigma_finite_measure T by fact
hoelzl@47694:   interpret ST: pair_sigma_finite S T ..
hoelzl@47694: 
hoelzl@47694:   have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)"
hoelzl@49786:     by (rule entropy_distr[OF Py])
wenzelm@53015:   also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@47694:     using b_gt_1 Py[THEN distributed_real_measurable]
hoelzl@47694:     by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong)
wenzelm@53015:   finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^sub>M T))" .
hoelzl@49791: 
wenzelm@53015:   have ae2: "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
wenzelm@53015:   moreover have ae4: "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Py (snd x)"
hoelzl@47694:     using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
hoelzl@49788:   moreover note ae5 = Pxy[THEN distributed_real_AE]
wenzelm@53015:   ultimately have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Pxy x \<and> 0 \<le> Py (snd x) \<and>
hoelzl@49790:     (Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Py (snd x)))"
hoelzl@47694:     by eventually_elim auto
wenzelm@53015:   then have ae: "AE x in S \<Otimes>\<^sub>M T.
hoelzl@47694:      Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))"
hoelzl@47694:     by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1)
hoelzl@49791:   have "conditional_entropy b S T X Y = 
wenzelm@53015:     - (\<integral>x. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@49791:     unfolding conditional_entropy_generic_eq[OF S T Py Pxy] neg_equal_iff_equal
hoelzl@49791:     apply (intro integral_cong_AE)
hoelzl@49791:     using ae
hoelzl@49791:     apply eventually_elim
hoelzl@49791:     apply auto
hoelzl@47694:     done
wenzelm@53015:   also have "\<dots> = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^sub>M T)) - - (\<integral>x.  Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@49791:     by (simp add: integral_diff[OF I1 I2])
hoelzl@49791:   finally show ?thesis 
hoelzl@49791:     unfolding conditional_entropy_generic_eq[OF S T Py Pxy] entropy_distr[OF Pxy] e_eq
hoelzl@49791:     by (simp add: split_beta')
hoelzl@49791: qed
hoelzl@49791: 
hoelzl@49791: lemma (in information_space) conditional_entropy_eq_entropy_simple:
hoelzl@49791:   assumes X: "simple_function M X" and Y: "simple_function M Y"
wenzelm@53015:   shows "\<H>(X | Y) = entropy b (count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x)) - \<H>(Y)"
hoelzl@49791: proof -
wenzelm@53015:   have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
hoelzl@49791:     (is "?P = ?C") using X Y by (simp add: simple_functionD pair_measure_count_space)
hoelzl@49791:   show ?thesis
hoelzl@49791:     by (rule conditional_entropy_eq_entropy sigma_finite_measure_count_space_finite
hoelzl@49791:                  simple_functionD  X Y simple_distributed simple_distributedI[OF _ refl]
hoelzl@49791:                  simple_distributed_joint simple_function_Pair integrable_count_space)+
hoelzl@49791:        (auto simp: `?P = ?C` intro!: integrable_count_space simple_functionD  X Y)
hoelzl@39097: qed
hoelzl@39097: 
hoelzl@40859: lemma (in information_space) conditional_entropy_eq:
hoelzl@49792:   assumes Y: "simple_distributed M Y Py"
hoelzl@47694:   assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:     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: proof (subst conditional_entropy_generic_eq[OF _ _
hoelzl@49790:   simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@49792:   have "finite ((\<lambda>x. (X x, Y x))`space M)"
hoelzl@49792:     using XY unfolding simple_distributed_def by auto
hoelzl@49792:   from finite_imageI[OF this, of fst]
hoelzl@49792:   have [simp]: "finite (X`space M)"
hoelzl@49792:     by (simp add: image_compose[symmetric] comp_def)
hoelzl@47694:   note Y[THEN simple_distributed_finite, simp]
hoelzl@47694:   show "sigma_finite_measure (count_space (X ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   show "sigma_finite_measure (count_space (Y ` space M))"
hoelzl@47694:     by (simp add: sigma_finite_measure_count_space_finite)
hoelzl@47694:   let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
wenzelm@53015:   have "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
hoelzl@47694:     (is "?P = ?C")
hoelzl@49792:     using Y by (simp add: simple_distributed_finite pair_measure_count_space)
hoelzl@47694:   have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) =
hoelzl@47694:     (\<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:     by auto
hoelzl@49792:   from Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) =
hoelzl@47694:     - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
hoelzl@47694:     by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta')
hoelzl@47694: qed
hoelzl@39097: 
hoelzl@47694: lemma (in information_space) conditional_mutual_information_eq_conditional_entropy:
hoelzl@41689:   assumes X: "simple_function M X" and Y: "simple_function M Y"
hoelzl@47694:   shows "\<I>(X ; X | Y) = \<H>(X | Y)"
hoelzl@47694: proof -
hoelzl@47694:   def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0"
hoelzl@47694:   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:   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:   let ?M = "X`space M \<times> X`space M \<times> Y`space M"
hoelzl@39097: 
hoelzl@47694:   note XY = simple_function_Pair[OF X Y]
hoelzl@47694:   note XXY = simple_function_Pair[OF X XY]
hoelzl@47694:   have Py: "simple_distributed M Y Py"
hoelzl@47694:     using Y by (rule simple_distributedI) (auto simp: Py_def)
hoelzl@47694:   have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@47694:     using XY by (rule simple_distributedI) (auto simp: Pxy_def)
hoelzl@47694:   have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy"
hoelzl@47694:     using XXY by (rule simple_distributedI) (auto simp: Pxxy_def)
hoelzl@47694:   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:     by auto
hoelzl@47694:   have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A"
hoelzl@47694:     by (auto simp: inj_on_def)
hoelzl@47694:   have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)"
hoelzl@47694:     by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob])
hoelzl@47694:   have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair)
hoelzl@47694:   then show ?thesis
hoelzl@47694:     apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy])
hoelzl@49792:     apply (subst conditional_entropy_eq[OF Py Pxy])
hoelzl@47694:     apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj]
hoelzl@47694:                 log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space)
hoelzl@47694:     using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE]
hoelzl@49790:   apply (auto simp add: not_le[symmetric] AE_count_space)
hoelzl@47694:     done
hoelzl@47694: qed
hoelzl@47694: 
hoelzl@47694: lemma (in information_space) conditional_entropy_nonneg:
hoelzl@47694:   assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)"
hoelzl@47694:   using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y]
hoelzl@47694:   by simp
hoelzl@36080: 
hoelzl@39097: subsection {* Equalities *}
hoelzl@39097: 
hoelzl@47694: lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
hoelzl@47694:   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@47694:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@47694:   assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015:   assumes Ix: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
wenzelm@53015:   assumes Iy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
wenzelm@53015:   assumes Ixy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
wenzelm@53015:   shows  "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
hoelzl@40859: proof -
wenzelm@53015:   have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@47694:     using b_gt_1 Px[THEN distributed_real_measurable]
hoelzl@49786:     apply (subst entropy_distr[OF Px])
hoelzl@47694:     apply (subst distributed_transform_integral[OF Pxy Px, where T=fst])
hoelzl@47694:     apply (auto intro!: integral_cong)
hoelzl@47694:     done
hoelzl@47694: 
wenzelm@53015:   have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@47694:     using b_gt_1 Py[THEN distributed_real_measurable]
hoelzl@49786:     apply (subst entropy_distr[OF Py])
hoelzl@47694:     apply (subst distributed_transform_integral[OF Pxy Py, where T=snd])
hoelzl@47694:     apply (auto intro!: integral_cong)
hoelzl@47694:     done
hoelzl@47694: 
hoelzl@47694:   interpret S: sigma_finite_measure S by fact
hoelzl@47694:   interpret T: sigma_finite_measure T by fact
hoelzl@47694:   interpret ST: pair_sigma_finite S T ..
wenzelm@53015:   have ST: "sigma_finite_measure (S \<Otimes>\<^sub>M T)" ..
hoelzl@47694: 
wenzelm@53015:   have XY: "entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^sub>M T))"
hoelzl@49786:     by (subst entropy_distr[OF Pxy]) (auto intro!: integral_cong)
hoelzl@47694:   
wenzelm@53015:   have "AE x in S \<Otimes>\<^sub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair)
wenzelm@53015:   moreover have "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
hoelzl@47694:     by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
wenzelm@53015:   moreover have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Px (fst x)"
hoelzl@47694:     using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
wenzelm@53015:   moreover have "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Py (snd x)"
hoelzl@47694:     using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
hoelzl@47694:   moreover note Pxy[THEN distributed_real_AE]
wenzelm@53015:   ultimately have "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) = 
hoelzl@47694:     Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
hoelzl@47694:     (is "AE x in _. ?f x = ?g x")
hoelzl@47694:   proof eventually_elim
hoelzl@47694:     case (goal1 x)
hoelzl@47694:     show ?case
hoelzl@47694:     proof cases
hoelzl@47694:       assume "Pxy x \<noteq> 0"
hoelzl@47694:       with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x"
hoelzl@47694:         by auto
hoelzl@47694:       then show ?thesis
hoelzl@47694:         using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
hoelzl@47694:     qed simp
hoelzl@47694:   qed
hoelzl@47694: 
wenzelm@53015:   have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?f"
hoelzl@47694:     unfolding X Y XY
hoelzl@47694:     apply (subst integral_diff)
hoelzl@47694:     apply (intro integral_diff Ixy Ix Iy)+
hoelzl@47694:     apply (subst integral_diff)
hoelzl@47694:     apply (intro integral_diff Ixy Ix Iy)+
hoelzl@47694:     apply (simp add: field_simps)
hoelzl@47694:     done
wenzelm@53015:   also have "\<dots> = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?g"
hoelzl@47694:     using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE)
hoelzl@47694:   also have "\<dots> = mutual_information b S T X Y"
hoelzl@47694:     by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
hoelzl@47694:   finally show ?thesis ..
hoelzl@47694: qed
hoelzl@47694: 
hoelzl@49802: lemma (in information_space) mutual_information_eq_entropy_conditional_entropy':
hoelzl@49802:   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@49802:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49802:   assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
wenzelm@53015:   assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wenzelm@53015:   assumes Ix: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
wenzelm@53015:   assumes Iy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
wenzelm@53015:   assumes Ixy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
hoelzl@49802:   shows  "mutual_information b S T X Y = entropy b S X - conditional_entropy b S T X Y"
hoelzl@49802:   using
hoelzl@49802:     mutual_information_eq_entropy_conditional_entropy_distr[OF S T Px Py Pxy Ix Iy Ixy]
hoelzl@49802:     conditional_entropy_eq_entropy[OF S T Py Pxy Ixy Iy]
hoelzl@49802:   by simp
hoelzl@49802: 
hoelzl@47694: lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
hoelzl@47694:   assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
hoelzl@47694:   shows  "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"
hoelzl@47694: proof -
hoelzl@47694:   have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
hoelzl@47694:     using sf_X by (rule simple_distributedI) auto
hoelzl@47694:   have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
hoelzl@47694:     using sf_Y by (rule simple_distributedI) auto
hoelzl@47694:   have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694:     using sf_X sf_Y by (rule simple_function_Pair)
hoelzl@47694:   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:     by (rule simple_distributedI) auto
hoelzl@47694:   from simple_distributed_joint_finite[OF this, simp]
wenzelm@53015:   have eq: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
hoelzl@47694:     by (simp add: pair_measure_count_space)
hoelzl@47694: 
wenzelm@53015:   have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^sub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))"
hoelzl@47694:     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:     by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space)
hoelzl@47694:   then show ?thesis
hoelzl@49791:     unfolding conditional_entropy_eq_entropy_simple[OF sf_X sf_Y] by simp
hoelzl@47694: qed
hoelzl@47694: 
hoelzl@47694: lemma (in information_space) mutual_information_nonneg_simple:
hoelzl@47694:   assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
hoelzl@47694:   shows  "0 \<le> \<I>(X ; Y)"
hoelzl@47694: proof -
hoelzl@47694:   have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
hoelzl@47694:     using sf_X by (rule simple_distributedI) auto
hoelzl@47694:   have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
hoelzl@47694:     using sf_Y by (rule simple_distributedI) auto
hoelzl@47694: 
hoelzl@47694:   have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
hoelzl@47694:     using sf_X sf_Y by (rule simple_function_Pair)
hoelzl@47694:   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:     by (rule simple_distributedI) auto
hoelzl@47694: 
hoelzl@47694:   from simple_distributed_joint_finite[OF this, simp]
wenzelm@53015:   have eq: "count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"
hoelzl@47694:     by (simp add: pair_measure_count_space)
hoelzl@47694: 
hoelzl@40859:   show ?thesis
hoelzl@47694:     by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
hoelzl@47694:        (simp_all add: eq integrable_count_space sigma_finite_measure_count_space_finite)
hoelzl@40859: qed
hoelzl@36080: 
hoelzl@40859: lemma (in information_space) conditional_entropy_less_eq_entropy:
hoelzl@41689:   assumes X: "simple_function M X" and Z: "simple_function M Z"
hoelzl@40859:   shows "\<H>(X | Z) \<le> \<H>(X)"
hoelzl@36624: proof -
hoelzl@47694:   have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple)
hoelzl@47694:   also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
hoelzl@47694:   finally show ?thesis by auto
hoelzl@36080: qed
hoelzl@36080: 
hoelzl@49803: lemma (in information_space) 
hoelzl@49803:   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
hoelzl@49803:   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
hoelzl@49803:   assumes Px: "finite_entropy S X Px" and Py: "finite_entropy T Y Py"
wenzelm@53015:   assumes Pxy: "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
hoelzl@49803:   shows "conditional_entropy b S T X Y \<le> entropy b S X"
hoelzl@49803: proof -
hoelzl@49803: 
hoelzl@49803:   have "0 \<le> mutual_information b S T X Y" 
hoelzl@49803:     by (rule mutual_information_nonneg') fact+
hoelzl@49803:   also have "\<dots> = entropy b S X - conditional_entropy b S T X Y"
hoelzl@49803:     apply (rule mutual_information_eq_entropy_conditional_entropy')
hoelzl@49803:     using assms
hoelzl@49803:     by (auto intro!: finite_entropy_integrable finite_entropy_distributed
hoelzl@49803:       finite_entropy_integrable_transform[OF Px]
hoelzl@49803:       finite_entropy_integrable_transform[OF Py])
hoelzl@49803:   finally show ?thesis by auto
hoelzl@49803: qed
hoelzl@49803: 
hoelzl@40859: lemma (in information_space) entropy_chain_rule:
hoelzl@41689:   assumes X: "simple_function M X" and Y: "simple_function M Y"
hoelzl@40859:   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
hoelzl@40859: proof -
hoelzl@47694:   note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl]
hoelzl@47694:   note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl]
hoelzl@47694:   note simple_distributed_joint_finite[OF this, simp]
hoelzl@47694:   let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)"
hoelzl@47694:   let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)"
hoelzl@47694:   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:   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:     using XY by (rule entropy_simple_distributed)
hoelzl@47694:   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:     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:   also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))"
hoelzl@47694:     by (auto intro!: setsum_cong)
wenzelm@53015:   also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^sub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))"
hoelzl@49786:     by (subst entropy_distr[OF simple_distributed_joint[OF YX]])
hoelzl@47694:        (auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite
hoelzl@47694:              cong del: setsum_cong  intro!: setsum_mono_zero_left)
wenzelm@53015:   finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^sub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" .
hoelzl@47694:   then show ?thesis
hoelzl@49791:     unfolding conditional_entropy_eq_entropy_simple[OF Y X] by simp
hoelzl@36624: qed
hoelzl@36624: 
hoelzl@40859: lemma (in information_space) entropy_partition:
hoelzl@47694:   assumes X: "simple_function M X"
hoelzl@47694:   shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)"
hoelzl@36624: proof -
hoelzl@47694:   note fX = simple_function_compose[OF X, of f]  
hoelzl@47694:   have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto
hoelzl@47694:   have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A"
hoelzl@47694:     by (auto simp: inj_on_def)
hoelzl@47694:   show ?thesis
hoelzl@47694:     apply (subst entropy_chain_rule[symmetric, OF fX X])
hoelzl@47694:     apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]])
hoelzl@47694:     apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
hoelzl@47694:     unfolding eq
hoelzl@47694:     apply (subst setsum_reindex[OF inj])
hoelzl@47694:     apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
hoelzl@47694:     done
hoelzl@36624: qed
hoelzl@36624: 
hoelzl@40859: corollary (in information_space) entropy_data_processing:
hoelzl@41689:   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
hoelzl@40859: proof -
hoelzl@47694:   note fX = simple_function_compose[OF X, of f]
hoelzl@47694:   from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
hoelzl@40859:   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
hoelzl@47694:     by (auto intro: conditional_entropy_nonneg[OF X fX])
hoelzl@40859: qed
hoelzl@36624: 
hoelzl@40859: corollary (in information_space) entropy_of_inj:
hoelzl@41689:   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
hoelzl@36624:   shows "\<H>(f \<circ> X) = \<H>(X)"
hoelzl@36624: proof (rule antisym)
hoelzl@40859:   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
hoelzl@36624: next
hoelzl@41689:   have sf: "simple_function M (f \<circ> X)"
hoelzl@40859:     using X by auto
hoelzl@36624:   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
hoelzl@47694:     unfolding o_assoc
hoelzl@47694:     apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
hoelzl@47694:     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:     apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def)
hoelzl@47694:     done
hoelzl@36624:   also have "... \<le> \<H>(f \<circ> X)"
hoelzl@40859:     using entropy_data_processing[OF sf] .
hoelzl@36624:   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
hoelzl@36624: qed
hoelzl@36624: 
hoelzl@36080: end




