--- a/src/HOL/Probability/Information.thy Thu Jun 09 13:55:11 2011 +0200
+++ b/src/HOL/Probability/Information.thy Thu Jun 09 14:04:34 2011 +0200
@@ -7,14 +7,10 @@
theory Information
imports
- Probability_Measure
+ Independent_Family
"~~/src/HOL/Library/Convex"
begin
-lemma (in prob_space) not_zero_less_distribution[simp]:
- "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
- using distribution_positive[of X A] by arith
-
lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
by (subst log_le_cancel_iff) auto
@@ -175,7 +171,211 @@
Kullback$-$Leibler distance. *}
definition
- "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv M \<nu> x)) \<partial>M\<lparr>measure := \<nu>\<rparr>"
+ "entropy_density b M \<nu> = log b \<circ> real \<circ> RN_deriv M \<nu>"
+
+definition
+ "KL_divergence b M \<nu> = integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) (entropy_density b M \<nu>)"
+
+lemma (in information_space) measurable_entropy_density:
+ assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
+ assumes ac: "absolutely_continuous \<nu>"
+ shows "entropy_density b M \<nu> \<in> borel_measurable M"
+proof -
+ interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
+ have "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by fact
+ from RN_deriv[OF this ac] b_gt_1 show ?thesis
+ unfolding entropy_density_def
+ by (intro measurable_comp) auto
+qed
+
+lemma (in information_space) KL_gt_0:
+ assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
+ assumes ac: "absolutely_continuous \<nu>"
+ assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
+ assumes A: "A \<in> sets M" "\<nu> A \<noteq> \<mu> A"
+ shows "0 < KL_divergence b M \<nu>"
+proof -
+ interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
+ have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
+ have fms: "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" by default
+ note RN = RN_deriv[OF ms ac]
+
+ from real_RN_deriv[OF fms ac] guess D . note D = this
+ with absolutely_continuous_AE[OF ms] ac
+ have D\<nu>: "AE x in M\<lparr>measure := \<nu>\<rparr>. RN_deriv M \<nu> x = extreal (D x)"
+ by auto
+
+ def f \<equiv> "\<lambda>x. if D x = 0 then 1 else 1 / D x"
+ with D have f_borel: "f \<in> borel_measurable M"
+ by (auto intro!: measurable_If)
+
+ have "KL_divergence b M \<nu> = 1 / ln b * (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
+ unfolding KL_divergence_def using int b_gt_1
+ by (simp add: integral_cmult)
+
+ { fix A assume "A \<in> sets M"
+ with RN D have "\<nu>.\<mu> A = (\<integral>\<^isup>+ x. extreal (D x) * indicator A x \<partial>M)"
+ by (auto intro!: positive_integral_cong_AE) }
+ note D_density = this
+
+ have ln_entropy: "(\<lambda>x. ln b * entropy_density b M \<nu> x) \<in> borel_measurable M"
+ using measurable_entropy_density[OF ps ac] by auto
+
+ have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x)"
+ using int by auto
+ moreover have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x) \<longleftrightarrow>
+ integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
+ using D D_density ln_entropy
+ by (intro integral_translated_density) auto
+ ultimately have M_int: "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
+ by simp
+
+ have D_neg: "(\<integral>\<^isup>+ x. extreal (- D x) \<partial>M) = 0"
+ using D by (subst positive_integral_0_iff_AE) auto
+
+ have "(\<integral>\<^isup>+ x. extreal (D x) \<partial>M) = \<nu> (space M)"
+ using RN D by (auto intro!: positive_integral_cong_AE)
+ then have D_pos: "(\<integral>\<^isup>+ x. extreal (D x) \<partial>M) = 1"
+ using \<nu>.measure_space_1 by simp
+
+ have "integrable M D"
+ using D_pos D_neg D by (auto simp: integrable_def)
+
+ have "integral\<^isup>L M D = 1"
+ using D_pos D_neg by (auto simp: lebesgue_integral_def)
+
+ let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
+ have [simp, intro]: "?D_set \<in> sets M"
+ using D by (auto intro: sets_Collect)
+
+ have "0 \<le> 1 - \<mu>' ?D_set"
+ using prob_le_1 by (auto simp: field_simps)
+ also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
+ using `integrable M D` `integral\<^isup>L M D = 1`
+ by (simp add: \<mu>'_def)
+ also have "\<dots> < (\<integral> x. D x * (ln b * entropy_density b M \<nu> x) \<partial>M)"
+ proof (rule integral_less_AE)
+ show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
+ using `integrable M D`
+ by (intro integral_diff integral_indicator) auto
+ next
+ show "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
+ by fact
+ next
+ show "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
+ proof
+ assume eq_0: "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
+ then have disj: "AE x. D x = 1 \<or> D x = 0"
+ using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect)
+
+ have "\<mu> {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
+ using D(1) by auto
+ also have "\<dots> = (\<integral>\<^isup>+ x. extreal (D x) * indicator {x\<in>space M. D x \<noteq> 0} x \<partial>M)"
+ using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_extreal_def)
+ also have "\<dots> = \<nu> {x\<in>space M. D x \<noteq> 0}"
+ using D(1) D_density by auto
+ also have "\<dots> = \<nu> (space M)"
+ using D_density D(1) by (auto intro!: positive_integral_cong simp: indicator_def)
+ finally have "AE x. D x = 1"
+ using D(1) \<nu>.measure_space_1 by (intro AE_I_eq_1) auto
+ then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. extreal (D x) * indicator A x\<partial>M)"
+ by (intro positive_integral_cong_AE) (auto simp: one_extreal_def[symmetric])
+ also have "\<dots> = \<nu> A"
+ using `A \<in> sets M` D_density by simp
+ finally show False using `A \<in> sets M` `\<nu> A \<noteq> \<mu> A` by simp
+ qed
+ show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
+ using D(1) by (auto intro: sets_Collect)
+
+ show "AE t. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
+ D t - indicator ?D_set t \<noteq> D t * (ln b * entropy_density b M \<nu> t)"
+ using D(2)
+ proof (elim AE_mp, safe intro!: AE_I2)
+ fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0"
+ and RN: "RN_deriv M \<nu> t = extreal (D t)"
+ and eq: "D t - indicator ?D_set t = D t * (ln b * entropy_density b M \<nu> t)"
+
+ have "D t - 1 = D t - indicator ?D_set t"
+ using Dt by simp
+ also note eq
+ also have "D t * (ln b * entropy_density b M \<nu> t) = - D t * ln (1 / D t)"
+ using RN b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
+ by (simp add: entropy_density_def log_def ln_div less_le)
+ finally have "ln (1 / D t) = 1 / D t - 1"
+ using `D t \<noteq> 0` by (auto simp: field_simps)
+ from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
+ show False by auto
+ qed
+
+ show "AE t. D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
+ using D(2)
+ proof (elim AE_mp, intro AE_I2 impI)
+ fix t assume "t \<in> space M" and RN: "RN_deriv M \<nu> t = extreal (D t)"
+ show "D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
+ proof cases
+ assume asm: "D t \<noteq> 0"
+ then have "0 < D t" using `0 \<le> D t` by auto
+ then have "0 < 1 / D t" by auto
+ have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
+ using asm `t \<in> space M` by (simp add: field_simps)
+ also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
+ using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
+ also have "\<dots> = D t * (ln b * entropy_density b M \<nu> t)"
+ using `0 < D t` RN b_gt_1
+ by (simp_all add: log_def ln_div entropy_density_def)
+ finally show ?thesis by simp
+ qed simp
+ qed
+ qed
+ also have "\<dots> = (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
+ using D D_density ln_entropy
+ by (intro integral_translated_density[symmetric]) auto
+ also have "\<dots> = ln b * (\<integral> x. entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
+ using int by (rule \<nu>.integral_cmult)
+ finally show "0 < KL_divergence b M \<nu>"
+ using b_gt_1 by (auto simp: KL_divergence_def zero_less_mult_iff)
+qed
+
+lemma (in sigma_finite_measure) KL_eq_0:
+ assumes eq: "\<forall>A\<in>sets M. \<nu> A = measure M A"
+ shows "KL_divergence b M \<nu> = 0"
+proof -
+ have "AE x. 1 = RN_deriv M \<nu> x"
+ proof (rule RN_deriv_unique)
+ show "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
+ using eq by (intro measure_space_cong) auto
+ show "absolutely_continuous \<nu>"
+ unfolding absolutely_continuous_def using eq by auto
+ show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x. 0 \<le> (1 :: extreal)" by auto
+ fix A assume "A \<in> sets M"
+ with eq show "\<nu> A = \<integral>\<^isup>+ x. 1 * indicator A x \<partial>M" by simp
+ qed
+ then have "AE x. log b (real (RN_deriv M \<nu> x)) = 0"
+ by (elim AE_mp) simp
+ from integral_cong_AE[OF this]
+ have "integral\<^isup>L M (entropy_density b M \<nu>) = 0"
+ by (simp add: entropy_density_def comp_def)
+ with eq show "KL_divergence b M \<nu> = 0"
+ unfolding KL_divergence_def
+ by (subst integral_cong_measure) auto
+qed
+
+lemma (in information_space) KL_eq_0_imp:
+ assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
+ assumes ac: "absolutely_continuous \<nu>"
+ assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
+ assumes KL: "KL_divergence b M \<nu> = 0"
+ shows "\<forall>A\<in>sets M. \<nu> A = \<mu> A"
+ by (metis less_imp_neq KL_gt_0 assms)
+
+lemma (in information_space) KL_ge_0:
+ assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
+ assumes ac: "absolutely_continuous \<nu>"
+ assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
+ shows "0 \<le> KL_divergence b M \<nu>"
+ using KL_eq_0 KL_gt_0[OF ps ac int]
+ by (cases "\<forall>A\<in>sets M. \<nu> A = measure M A") (auto simp: le_less)
+
lemma (in sigma_finite_measure) KL_divergence_vimage:
assumes T: "T \<in> measure_preserving M M'"
@@ -209,7 +409,7 @@
have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
show ?thesis
- unfolding KL_divergence_def
+ unfolding KL_divergence_def entropy_density_def comp_def
proof (subst \<nu>'.integral_vimage[OF sa T'])
show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
@@ -233,9 +433,9 @@
proof -
interpret \<nu>: measure_space ?\<nu> by fact
have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
- by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def)
+ by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def entropy_density_def)
also have "\<dots> = KL_divergence b N \<nu>'"
- by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
+ by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def entropy_density_def comp_def)
finally show ?thesis .
qed
@@ -243,7 +443,7 @@
assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
assumes ac: "absolutely_continuous \<nu>"
shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
-proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
+proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v] entropy_density_def)
interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
@@ -257,27 +457,10 @@
and "1 < b"
shows "0 \<le> KL_divergence b M \<nu>"
proof -
+ interpret information_space M by default fact
interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have ms: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
-
- have "- (KL_divergence b M \<nu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
- proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
- show "finite (space M)" using finite_space by simp
- show "1 < b" by fact
- show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
- using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
-
- fix x assume "x \<in> space M"
- then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
- { assume "0 < real (\<nu> {x})"
- then have "\<nu> {x} \<noteq> 0" by auto
- then have "\<mu> {x} \<noteq> 0"
- using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
- thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
- show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
- using real_measure[OF x] v.real_measure[of "{x}"] x by auto
- qed
- thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
+ have ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)" by default
+ from KL_ge_0[OF this ac v.integral_finite_singleton(1)] show ?thesis .
qed
subsection {* Mutual Information *}
@@ -287,6 +470,163 @@
KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
(extreal\<circ>joint_distribution X Y)"
+lemma (in information_space)
+ fixes S T X Y
+ defines "P \<equiv> S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
+ shows "indep_var S X T Y \<longleftrightarrow>
+ (random_variable S X \<and> random_variable T Y \<and>
+ measure_space.absolutely_continuous P (extreal\<circ>joint_distribution X Y) \<and>
+ integrable (P\<lparr>measure := (extreal\<circ>joint_distribution X Y)\<rparr>)
+ (entropy_density b P (extreal\<circ>joint_distribution X Y)) \<and>
+ mutual_information b S T X Y = 0)"
+proof safe
+ assume indep: "indep_var S X T Y"
+ then have "random_variable S X" "random_variable T Y"
+ by (blast dest: indep_var_rv1 indep_var_rv2)+
+ then show "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
+ by blast+
+
+ interpret X: prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
+ by (rule distribution_prob_space) fact
+ interpret Y: prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
+ by (rule distribution_prob_space) fact
+ interpret XY: pair_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
+ interpret XY: information_space XY.P b by default (rule b_gt_1)
+
+ let ?J = "XY.P\<lparr> measure := (extreal\<circ>joint_distribution X Y) \<rparr>"
+ { fix A assume "A \<in> sets XY.P"
+ then have "extreal (joint_distribution X Y A) = XY.\<mu> A"
+ using indep_var_distributionD[OF indep]
+ by (simp add: XY.P.finite_measure_eq) }
+ note j_eq = this
+
+ interpret J: prob_space ?J
+ using j_eq by (intro XY.prob_space_cong) auto
+
+ have ac: "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
+ by (simp add: XY.absolutely_continuous_def j_eq)
+ then show "measure_space.absolutely_continuous P (extreal\<circ>joint_distribution X Y)"
+ unfolding P_def .
+
+ have ed: "entropy_density b XY.P (extreal\<circ>joint_distribution X Y) \<in> borel_measurable XY.P"
+ by (rule XY.measurable_entropy_density) (default | fact)+
+
+ have "AE x in XY.P. 1 = RN_deriv XY.P (extreal\<circ>joint_distribution X Y) x"
+ proof (rule XY.RN_deriv_unique[OF _ ac])
+ show "measure_space ?J" by default
+ fix A assume "A \<in> sets XY.P"
+ then show "(extreal\<circ>joint_distribution X Y) A = (\<integral>\<^isup>+ x. 1 * indicator A x \<partial>XY.P)"
+ by (simp add: j_eq)
+ qed (insert XY.measurable_const[of 1 borel], auto)
+ then have ae_XY: "AE x in XY.P. entropy_density b XY.P (extreal\<circ>joint_distribution X Y) x = 0"
+ by (elim XY.AE_mp) (simp add: entropy_density_def)
+ have ae_J: "AE x in ?J. entropy_density b XY.P (extreal\<circ>joint_distribution X Y) x = 0"
+ proof (rule XY.absolutely_continuous_AE)
+ show "measure_space ?J" by default
+ show "XY.absolutely_continuous (measure ?J)"
+ using ac by simp
+ qed (insert ae_XY, simp_all)
+ then show "integrable (P\<lparr>measure := (extreal\<circ>joint_distribution X Y)\<rparr>)
+ (entropy_density b P (extreal\<circ>joint_distribution X Y))"
+ unfolding P_def
+ using ed XY.measurable_const[of 0 borel]
+ by (subst J.integrable_cong_AE) auto
+
+ show "mutual_information b S T X Y = 0"
+ unfolding mutual_information_def KL_divergence_def P_def
+ by (subst J.integral_cong_AE[OF ae_J]) simp
+next
+ assume "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
+ then have rvs: "random_variable S X" "random_variable T Y" by blast+
+
+ interpret X: prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
+ by (rule distribution_prob_space) fact
+ interpret Y: prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
+ by (rule distribution_prob_space) fact
+ interpret XY: pair_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
+ interpret XY: information_space XY.P b by default (rule b_gt_1)
+
+ let ?J = "XY.P\<lparr> measure := (extreal\<circ>joint_distribution X Y) \<rparr>"
+ interpret J: prob_space ?J
+ using rvs by (intro joint_distribution_prob_space) auto
+
+ assume ac: "measure_space.absolutely_continuous P (extreal\<circ>joint_distribution X Y)"
+ assume int: "integrable (P\<lparr>measure := (extreal\<circ>joint_distribution X Y)\<rparr>)
+ (entropy_density b P (extreal\<circ>joint_distribution X Y))"
+ assume I_eq_0: "mutual_information b S T X Y = 0"
+
+ have eq: "\<forall>A\<in>sets XY.P. (extreal \<circ> joint_distribution X Y) A = XY.\<mu> A"
+ proof (rule XY.KL_eq_0_imp)
+ show "prob_space ?J" by default
+ show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
+ using ac by (simp add: P_def)
+ show "integrable ?J (entropy_density b XY.P (extreal\<circ>joint_distribution X Y))"
+ using int by (simp add: P_def)
+ show "KL_divergence b XY.P (extreal\<circ>joint_distribution X Y) = 0"
+ using I_eq_0 unfolding mutual_information_def by (simp add: P_def)
+ qed
+
+ { fix S X assume "sigma_algebra S"
+ interpret S: sigma_algebra S by fact
+ have "Int_stable \<lparr>space = space M, sets = {X -` A \<inter> space M |A. A \<in> sets S}\<rparr>"
+ proof (safe intro!: Int_stableI)
+ fix A B assume "A \<in> sets S" "B \<in> sets S"
+ then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
+ by (intro exI[of _ "A \<inter> B"]) auto
+ qed }
+ note Int_stable = this
+
+ show "indep_var S X T Y" unfolding indep_var_eq
+ proof (intro conjI indep_set_sigma_sets Int_stable)
+ show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
+ proof (safe intro!: indep_setI)
+ { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
+ using `X \<in> measurable M S` by (auto intro: measurable_sets) }
+ { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
+ using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
+ next
+ fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
+ have "extreal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) =
+ extreal (joint_distribution X Y (A \<times> B))"
+ unfolding distribution_def
+ by (intro arg_cong[where f="\<lambda>C. extreal (prob C)"]) auto
+ also have "\<dots> = XY.\<mu> (A \<times> B)"
+ using ab eq by (auto simp: XY.finite_measure_eq)
+ also have "\<dots> = extreal (distribution X A) * extreal (distribution Y B)"
+ using ab by (simp add: XY.pair_measure_times)
+ finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
+ prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
+ unfolding distribution_def by simp
+ qed
+ qed fact+
+qed
+
+lemma (in information_space) mutual_information_commute_generic:
+ assumes X: "random_variable S X" and Y: "random_variable T Y"
+ assumes ac: "measure_space.absolutely_continuous
+ (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
+ shows "mutual_information b S T X Y = mutual_information b T S Y X"
+proof -
+ let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
+ interpret S: prob_space ?S using X by (rule distribution_prob_space)
+ interpret T: prob_space ?T using Y by (rule distribution_prob_space)
+ interpret P: pair_prob_space ?S ?T ..
+ interpret Q: pair_prob_space ?T ?S ..
+ show ?thesis
+ unfolding mutual_information_def
+ proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
+ show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
+ (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
+ using X Y unfolding measurable_def
+ unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
+ by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
+ have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
+ using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
+ then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
+ unfolding prob_space_def by simp
+ qed auto
+qed
+
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
@@ -356,32 +696,6 @@
unfolding mutual_information_def .
qed
-lemma (in information_space) mutual_information_commute_generic:
- assumes X: "random_variable S X" and Y: "random_variable T Y"
- assumes ac: "measure_space.absolutely_continuous
- (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
- shows "mutual_information b S T X Y = mutual_information b T S Y X"
-proof -
- let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
- interpret S: prob_space ?S using X by (rule distribution_prob_space)
- interpret T: prob_space ?T using Y by (rule distribution_prob_space)
- interpret P: pair_prob_space ?S ?T ..
- interpret Q: pair_prob_space ?T ?S ..
- show ?thesis
- unfolding mutual_information_def
- proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
- show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
- (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
- using X Y unfolding measurable_def
- unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
- by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
- have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
- using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
- then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
- unfolding prob_space_def by simp
- qed auto
-qed
-
lemma (in information_space) mutual_information_commute:
assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
shows "mutual_information b S T X Y = mutual_information b T S Y X"