reworked Probability theory: measures are not type restricted to positive extended reals
theory Information
imports
Probability_Space
"~~/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
lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
by (subst log_less_cancel_iff) auto
lemma setsum_cartesian_product':
"(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
unfolding setsum_cartesian_product by simp
section "Convex theory"
lemma log_setsum:
assumes "finite s" "s \<noteq> {}"
assumes "b > 1"
assumes "(\<Sum> i \<in> s. a i) = 1"
assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
proof -
have "convex_on {0 <..} (\<lambda> x. - log b x)"
by (rule minus_log_convex[OF `b > 1`])
hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
qed
lemma log_setsum':
assumes "finite s" "s \<noteq> {}"
assumes "b > 1"
assumes "(\<Sum> i \<in> s. a i) = 1"
assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
"\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
proof -
have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
using assms by (auto intro!: setsum_mono_zero_cong_left)
moreover have "log b (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) \<ge> (\<Sum> i \<in> s - {i. a i = 0}. a i * log b (y i))"
proof (rule log_setsum)
have "setsum a (s - {i. a i = 0}) = setsum a s"
using assms(1) by (rule setsum_mono_zero_cong_left) auto
thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
"finite (s - {i. a i = 0})" using assms by simp_all
show "s - {i. a i = 0} \<noteq> {}"
proof
assume *: "s - {i. a i = 0} = {}"
hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
with sum_1 show False by simp
qed
fix i assume "i \<in> s - {i. a i = 0}"
hence "i \<in> s" "a i \<noteq> 0" by simp_all
thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
qed fact+
ultimately show ?thesis by simp
qed
lemma log_setsum_divide:
assumes "finite S" and "S \<noteq> {}" and "1 < b"
assumes "(\<Sum>x\<in>S. g x) = 1"
assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
proof -
have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
using `1 < b` by (subst log_le_cancel_iff) auto
have "- (\<Sum>x\<in>S. g x * log b (g x / f x)) = (\<Sum>x\<in>S. g x * log b (f x / g x))"
proof (unfold setsum_negf[symmetric], rule setsum_cong)
fix x assume x: "x \<in> S"
show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
proof (cases "g x = 0")
case False
with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
qed simp
qed rule
also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
proof (rule log_setsum')
fix x assume x: "x \<in> S" "0 < g x"
with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
qed fact+
also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
split: split_if_asm)
also have "... \<le> log b (\<Sum>x\<in>S. f x)"
proof (rule log_mono)
have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
proof (rule setsum_strict_mono)
show "finite (S - {x. g x = 0})" using `finite S` by simp
show "S - {x. g x = 0} \<noteq> {}"
proof
assume "S - {x. g x = 0} = {}"
hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
with `(\<Sum>x\<in>S. g x) = 1` show False by simp
qed
fix x assume "x \<in> S - {x. g x = 0}"
thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
qed
finally show "0 < ?sum" .
show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
using `finite S` pos by (auto intro!: setsum_mono2)
qed
finally show ?thesis .
qed
lemma split_pairs:
"((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
"(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
section "Information theory"
locale information_space = prob_space +
fixes b :: real assumes b_gt_1: "1 < b"
context information_space
begin
text {* Introduce some simplification rules for logarithm of base @{term b}. *}
lemma log_neg_const:
assumes "x \<le> 0"
shows "log b x = log b 0"
proof -
{ fix u :: real
have "x \<le> 0" by fact
also have "0 < exp u"
using exp_gt_zero .
finally have "exp u \<noteq> x"
by auto }
then show "log b x = log b 0"
by (simp add: log_def ln_def)
qed
lemma log_mult_eq:
"log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
by (auto simp: zero_less_mult_iff mult_le_0_iff)
lemma log_inverse_eq:
"log b (inverse B) = (if 0 < B then - log b B else log b 0)"
using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
lemma log_divide_eq:
"log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
by (auto simp: zero_less_mult_iff mult_le_0_iff)
lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
end
subsection "Kullback$-$Leibler divergence"
text {* The Kullback$-$Leibler divergence is also known as relative entropy or
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>"
lemma (in sigma_finite_measure) KL_divergence_vimage:
assumes T: "T \<in> measure_preserving M M'"
and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
and "1 < b"
shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
proof -
interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
then have saM': "sigma_algebra M'" by simp
then interpret M': measure_space M' by (rule measure_space_vimage) fact
have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
proof safe
fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
then have N': "T' -` N \<inter> space M' \<in> sets M'"
using T' by (auto simp: measurable_def measure_preserving_def)
have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
then have "measure M' (T' -` N \<inter> space M') = 0"
using measure_preservingD[OF T N'] N_0 by auto
with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
unfolding M'.absolutely_continuous_def measurable_def by auto
qed
have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
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
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`])
have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
(\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
using inv' by (auto intro!: \<nu>'.integral_cong)
also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
using M ac AE
by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
(auto elim!: AE_mp)
finally show "?l = ?r" .
qed
qed
lemma (in sigma_finite_measure) KL_divergence_cong:
assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
assumes [simp]: "sets N = sets M" "space N = space M"
"\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
"\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
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)
also have "\<dots> = KL_divergence b N \<nu>'"
by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
finally show ?thesis .
qed
lemma (in finite_measure_space) KL_divergence_eq_finite:
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])
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"
using RN_deriv_finite_measure[OF ms ac]
by (auto intro!: setsum_cong simp: field_simps)
qed
lemma (in finite_prob_space) KL_divergence_positive_finite:
assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
assumes ac: "absolutely_continuous \<nu>"
and "1 < b"
shows "0 \<le> KL_divergence b M \<nu>"
proof -
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)
qed
subsection {* Mutual Information *}
definition (in prob_space)
"mutual_information b S T X Y =
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)"
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
abbreviation (in information_space)
mutual_information_Pow ("\<I>'(_ ; _')") where
"\<I>(X ; Y) \<equiv> mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
lemma (in prob_space) finite_variables_absolutely_continuous:
assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
shows "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)"
proof -
interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using X by (rule distribution_finite_prob_space)
interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
using Y by (rule distribution_finite_prob_space)
interpret XY: pair_finite_prob_space
"S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
using assms by (auto intro!: joint_distribution_finite_prob_space)
note rv = assms[THEN finite_random_variableD]
show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
proof (rule XY.absolutely_continuousI)
show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
then obtain a b where "x = (a, b)"
and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
by (cases x) (auto simp: space_pair_measure)
with finite_distribution_order(5,6)[OF X Y]
show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
qed
qed
lemma (in information_space)
assumes MX: "finite_random_variable MX X"
assumes MY: "finite_random_variable MY Y"
shows mutual_information_generic_eq:
"mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
joint_distribution X Y {(x,y)} *
log b (joint_distribution X Y {(x,y)} /
(distribution X {x} * distribution Y {y})))"
(is ?sum)
and mutual_information_positive_generic:
"0 \<le> mutual_information b MX MY X Y" (is ?positive)
proof -
interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using MX by (rule distribution_finite_prob_space)
interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
using MY by (rule distribution_finite_prob_space)
interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
using assms by (auto intro!: joint_distribution_finite_prob_space)
have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
show ?sum
unfolding Let_def mutual_information_def
by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
(auto simp add: space_pair_measure setsum_cartesian_product')
show ?positive
using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
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"
unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
unfolding joint_distribution_commute_singleton[of X Y]
by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
lemma (in information_space) mutual_information_commute_simple:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<I>(X;Y) = \<I>(Y;X)"
by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
lemma (in information_space) mutual_information_eq:
assumes "simple_function M X" "simple_function M Y"
shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
(distribution X {x} * distribution Y {y})))"
using assms by (simp add: mutual_information_generic_eq)
lemma (in information_space) mutual_information_generic_cong:
assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
unfolding mutual_information_def using X Y
by (simp cong: distribution_cong)
lemma (in information_space) mutual_information_cong:
assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
shows "\<I>(X; Y) = \<I>(X'; Y')"
unfolding mutual_information_def using X Y
by (simp cong: distribution_cong image_cong)
lemma (in information_space) mutual_information_positive:
assumes "simple_function M X" "simple_function M Y"
shows "0 \<le> \<I>(X;Y)"
using assms by (simp add: mutual_information_positive_generic)
subsection {* Entropy *}
abbreviation (in information_space)
entropy_Pow ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
lemma (in information_space) entropy_generic_eq:
fixes X :: "'a \<Rightarrow> 'c"
assumes MX: "finite_random_variable MX X"
shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
proof -
interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using MX by (rule distribution_finite_prob_space)
let "?X x" = "distribution X {x}"
let "?XX x y" = "joint_distribution X X {(x, y)}"
{ fix x y :: 'c
{ assume "x \<noteq> y"
then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
(if x = y then - ?X y * log b (?X y) else 0)"
by (auto simp: log_simps zero_less_mult_iff) }
note remove_XX = this
show ?thesis
unfolding entropy_def mutual_information_generic_eq[OF MX MX]
unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
using MX.finite_space by (auto simp: setsum_cases)
qed
lemma (in information_space) entropy_eq:
assumes "simple_function M X"
shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
using assms by (simp add: entropy_generic_eq)
lemma (in information_space) entropy_positive:
"simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
unfolding entropy_def by (simp add: mutual_information_positive)
lemma (in information_space) entropy_certainty_eq_0:
assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
shows "\<H>(X) = 0"
proof -
let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
note simple_function_imp_finite_random_variable[OF `simple_function M X`]
from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
interpret X: finite_prob_space ?X by simp
have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
using X.measure_compl[of "{x}"] assms by auto
also have "\<dots> = 0" using X.prob_space assms by auto
finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
{ fix y assume *: "y \<in> X ` space M"
{ assume asm: "y \<noteq> x"
with * have "{y} \<subseteq> X ` space M - {x}" by auto
from X.measure_mono[OF this] X0 asm *
have "distribution X {y} = 0" by (auto intro: antisym) }
then have "distribution X {y} = (if x = y then 1 else 0)"
using assms by auto }
note fi = this
have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
qed
lemma (in information_space) entropy_le_card_not_0:
assumes X: "simple_function M X"
shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
proof -
let "?p x" = "distribution X {x}"
have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
unfolding entropy_eq[OF X] setsum_negf[symmetric]
by (auto intro!: setsum_cong simp: log_simps)
also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
by (intro log_setsum') (auto simp: simple_function_def)
also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
finally show ?thesis
using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
qed
lemma (in prob_space) measure'_translate:
assumes X: "random_variable S X" and A: "A \<in> sets S"
shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
proof -
interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
using distribution_prob_space[OF X] .
from A show "S.\<mu>' A = distribution X A"
unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
qed
lemma (in information_space) entropy_uniform_max:
assumes X: "simple_function M X"
assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
shows "\<H>(X) = log b (real (card (X ` space M)))"
proof -
let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
note frv = simple_function_imp_finite_random_variable[OF X]
from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
interpret X: finite_prob_space ?X by simp
note rv = finite_random_variableD[OF frv]
have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
using `simple_function M X` not_empty by (auto simp: simple_function_def)
{ fix x assume "x \<in> space ?X"
moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
proof (rule X.uniform_prob)
fix x y assume "x \<in> space ?X" "y \<in> space ?X"
with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
by (subst (1 2) measure'_translate[OF rv]) auto
qed
ultimately have "distribution X {x} = 1 / card (space ?X)"
by (subst (asm) measure'_translate[OF rv]) auto }
thus ?thesis
using not_empty X.finite_space b_gt_1 card_gt0
by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
qed
lemma (in information_space) entropy_le_card:
assumes "simple_function M X"
shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
proof cases
assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
moreover
have "0 < card (X`space M)"
using `simple_function M X` not_empty
by (auto simp: card_gt_0_iff simple_function_def)
then have "log b 1 \<le> log b (real (card (X`space M)))"
using b_gt_1 by (intro log_le) auto
ultimately show ?thesis using assms by (simp add: entropy_eq)
next
assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
(is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
note entropy_le_card_not_0[OF assms]
also have "log b (real ?A) \<le> log b (real ?B)"
using b_gt_1 False not_empty `?A \<le> ?B` assms
by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
finally show ?thesis .
qed
lemma (in information_space) entropy_commute:
assumes "simple_function M X" "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
proof -
have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
using assms by (auto intro: simple_function_Pair)
have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
by auto
have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
by (auto intro!: inj_onI)
show ?thesis
unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
by (simp add: joint_distribution_commute[of Y X] split_beta)
qed
lemma (in information_space) entropy_eq_cartesian_product:
assumes "simple_function M X" "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
proof -
have sf: "simple_function M (\<lambda>x. (X x, Y x))"
using assms by (auto intro: simple_function_Pair)
{ fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
then have "joint_distribution X Y {x} = 0"
unfolding distribution_def by auto }
then show ?thesis using sf assms
unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
qed
subsection {* Conditional Mutual Information *}
definition (in prob_space)
"conditional_mutual_information b MX MY MZ X Y Z \<equiv>
mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
mutual_information b MX MZ X Z"
abbreviation (in information_space)
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
\<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
X Y Z"
lemma (in information_space) conditional_mutual_information_generic_eq:
assumes MX: "finite_random_variable MX X"
and MY: "finite_random_variable MY Y"
and MZ: "finite_random_variable MZ Z"
shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
(joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
(is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
proof -
let ?X = "\<lambda>x. distribution X {x}"
note finite_var = MX MY MZ
note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
note XYZ = finite_random_variable_pairI[OF MX YZ]
note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
note order1 =
finite_distribution_order(5,6)[OF finite_var(1) YZ]
finite_distribution_order(5,6)[OF finite_var(1,3)]
note random_var = finite_var[THEN finite_random_variableD]
note finite = finite_var(1) YZ finite_var(3) XZ YZX
have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
\<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
using finite_distribution_order(6)[OF finite_var(2) ZX]
by auto
have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
(is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
proof (safe intro!: setsum_cong)
fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
show "?L x y z = ?R x y z"
proof cases
assume "?XYZ x y z \<noteq> 0"
with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
using order1 order2 by (auto simp: less_le)
with b_gt_1 show ?thesis
by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
qed simp
qed
also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
(\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
setsum_left_distrib[symmetric]
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
by (intro setsum_cong refl) (simp add: space_pair_measure)
also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
(\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
conditional_mutual_information b MX MY MZ X Y Z"
unfolding conditional_mutual_information_def
unfolding mutual_information_generic_eq[OF finite_var(1,3)]
unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
finally show ?thesis by simp
qed
lemma (in information_space) conditional_mutual_information_eq:
assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
(joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
simp
lemma (in information_space) conditional_mutual_information_eq_mutual_information:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
proof -
have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
have C: "simple_function M (\<lambda>x. ())" by auto
show ?thesis
unfolding conditional_mutual_information_eq[OF X Y C]
unfolding mutual_information_eq[OF X Y]
by (simp add: setsum_cartesian_product' distribution_remove_const)
qed
lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
unfolding distribution_def using prob_space by auto
lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
lemma (in prob_space) setsum_distribution:
assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
lemma (in prob_space) setsum_real_distribution:
fixes MX :: "('c, 'd) measure_space_scheme"
assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
by auto
lemma (in information_space) conditional_mutual_information_generic_positive:
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
proof (cases "space MX \<times> space MY \<times> space MZ = {}")
case True show ?thesis
unfolding conditional_mutual_information_generic_eq[OF assms] True
by simp
next
case False
let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
let ?dXZ = "joint_distribution X Z"
let ?dYZ = "joint_distribution Y Z"
let ?dX = "distribution X"
let ?dZ = "distribution Z"
let ?M = "space MX \<times> space MY \<times> space MZ"
note YZ = finite_random_variable_pairI[OF Y Z]
note XZ = finite_random_variable_pairI[OF X Z]
note ZX = finite_random_variable_pairI[OF Z X]
note YZ = finite_random_variable_pairI[OF Y Z]
note XYZ = finite_random_variable_pairI[OF X YZ]
note finite = Z YZ XZ XYZ
have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
\<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
using finite_distribution_order(6)[OF Y ZX]
by auto
note order = order
finite_distribution_order(5,6)[OF X YZ]
finite_distribution_order(5,6)[OF Y Z]
have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
unfolding split_beta'
proof (rule log_setsum_divide)
show "?M \<noteq> {}" using False by simp
show "1 < b" using b_gt_1 .
show "finite ?M" using assms
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
unfolding setsum_cartesian_product'
unfolding setsum_commute[of _ "space MY"]
unfolding setsum_commute[of _ "space MZ"]
by (simp_all add: space_pair_measure
setsum_joint_distribution_singleton[OF X YZ]
setsum_joint_distribution_singleton[OF Y Z]
setsum_distribution[OF Z])
fix x assume "x \<in> ?M"
let ?x = "(fst x, fst (snd x), snd (snd x))"
show "0 \<le> ?dXYZ {?x}"
"0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
assume *: "0 < ?dXYZ {?x}"
with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
qed
also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
apply (simp add: setsum_cartesian_product')
apply (subst setsum_commute)
apply (subst (2) setsum_commute)
by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
setsum_joint_distribution_singleton[OF X Z]
setsum_joint_distribution_singleton[OF Y Z]
intro!: setsum_cong)
also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
unfolding setsum_real_distribution[OF Z] by simp
finally show ?thesis by simp
qed
lemma (in information_space) conditional_mutual_information_positive:
assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
shows "0 \<le> \<I>(X;Y|Z)"
by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
subsection {* Conditional Entropy *}
definition (in prob_space)
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
abbreviation (in information_space)
conditional_entropy_Pow ("\<H>'(_ | _')") where
"\<H>(X | Y) \<equiv> conditional_entropy b
\<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
lemma (in information_space) conditional_entropy_positive:
"simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
lemma (in information_space) conditional_entropy_generic_eq:
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
assumes MX: "finite_random_variable MX X"
assumes MZ: "finite_random_variable MZ Z"
shows "conditional_entropy b MX MZ X Z =
- (\<Sum>(x, z)\<in>space MX \<times> space MZ.
joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
proof -
interpret MX: finite_sigma_algebra MX using MX by simp
interpret MZ: finite_sigma_algebra MZ using MZ by simp
let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
let "?XZ x z" = "joint_distribution X Z {(x, z)}"
let "?Z z" = "distribution Z {z}"
let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
{ fix x z have "?XXZ x x z = ?XZ x z"
unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
note this[simp]
{ fix x x' :: 'c and z assume "x' \<noteq> x"
then have "?XXZ x x' z = 0"
by (auto simp: distribution_def empty_measure'[symmetric]
simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
note this[simp]
{ fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
= (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
by (auto intro!: setsum_cong)
also have "\<dots> = ?XZ x z * ?f x x z"
using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
using finite_distribution_order(6)[OF MX MZ]
by (auto simp: log_simps field_simps zero_less_mult_iff)
finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
note * = this
show ?thesis
unfolding conditional_entropy_def
unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
setsum_commute[of _ "space MZ"] *
intro!: setsum_cong)
qed
lemma (in information_space) conditional_entropy_eq:
assumes "simple_function M X" "simple_function M Z"
shows "\<H>(X | Z) =
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
joint_distribution X Z {(x, z)} *
log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
simp
lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(X | Y) =
-(\<Sum>y\<in>Y`space M. distribution Y {y} *
(\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
unfolding conditional_entropy_eq[OF assms]
using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib
intro!: setsum_cong)
lemma (in information_space) conditional_entropy_eq_cartesian_product:
assumes "simple_function M X" "simple_function M Y"
shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
joint_distribution X Y {(x,y)} *
log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
unfolding conditional_entropy_eq[OF assms]
by (auto intro!: setsum_cong simp: setsum_cartesian_product')
subsection {* Equalities *}
lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
assumes X: "simple_function M X" and Z: "simple_function M Z"
shows "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
proof -
let "?XZ x z" = "joint_distribution X Z {(x, z)}"
let "?Z z" = "distribution Z {z}"
let "?X x" = "distribution X {x}"
note fX = X[THEN simple_function_imp_finite_random_variable]
note fZ = Z[THEN simple_function_imp_finite_random_variable]
note finite_distribution_order[OF fX fZ, simp]
{ fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
note * = this
show ?thesis
unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
setsum_distribution)
qed
lemma (in information_space) conditional_entropy_less_eq_entropy:
assumes X: "simple_function M X" and Z: "simple_function M Z"
shows "\<H>(X | Z) \<le> \<H>(X)"
proof -
have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
with mutual_information_positive[OF X Z] entropy_positive[OF X]
show ?thesis by auto
qed
lemma (in information_space) entropy_chain_rule:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
proof -
let "?XY x y" = "joint_distribution X Y {(x, y)}"
let "?Y y" = "distribution Y {y}"
let "?X x" = "distribution X {x}"
note fX = X[THEN simple_function_imp_finite_random_variable]
note fY = Y[THEN simple_function_imp_finite_random_variable]
note finite_distribution_order[OF fX fY, simp]
{ fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
have "?XY x y * log b (?XY x y / ?X x) =
?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
note * = this
show ?thesis
using setsum_joint_distribution_singleton[OF fY fX]
unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
qed
section {* Partitioning *}
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
lemma subvimageI:
assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
shows "subvimage A f g"
using assms unfolding subvimage_def by blast
lemma subvimageE[consumes 1]:
assumes "subvimage A f g"
obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
using assms unfolding subvimage_def by blast
lemma subvimageD:
"\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
using assms unfolding subvimage_def by blast
lemma subvimage_subset:
"\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
unfolding subvimage_def by auto
lemma subvimage_idem[intro]: "subvimage A g g"
by (safe intro!: subvimageI)
lemma subvimage_comp_finer[intro]:
assumes svi: "subvimage A g h"
shows "subvimage A g (f \<circ> h)"
proof (rule subvimageI, simp)
fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
from svi[THEN subvimageD, OF this]
show "f (h x) = f (h y)" by simp
qed
lemma subvimage_comp_gran:
assumes svi: "subvimage A g h"
assumes inj: "inj_on f (g ` A)"
shows "subvimage A (f \<circ> g) h"
by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
lemma subvimage_comp:
assumes svi: "subvimage (f ` A) g h"
shows "subvimage A (g \<circ> f) (h \<circ> f)"
by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
lemma subvimage_trans:
assumes fg: "subvimage A f g"
assumes gh: "subvimage A g h"
shows "subvimage A f h"
by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
lemma subvimage_translator:
assumes svi: "subvimage A f g"
shows "\<exists>h. \<forall>x \<in> A. h (f x) = g x"
proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
fix x assume "x \<in> A"
show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
by (rule theI2[of _ "g x"])
(insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
qed
lemma subvimage_translator_image:
assumes svi: "subvimage A f g"
shows "\<exists>h. h ` f ` A = g ` A"
proof -
from subvimage_translator[OF svi]
obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
thus ?thesis
by (auto intro!: exI[of _ h]
simp: image_compose[symmetric] comp_def cong: image_cong)
qed
lemma subvimage_finite:
assumes svi: "subvimage A f g" and fin: "finite (f`A)"
shows "finite (g`A)"
proof -
from subvimage_translator_image[OF svi]
obtain h where "g`A = h`f`A" by fastsimp
with fin show "finite (g`A)" by simp
qed
lemma subvimage_disj:
assumes svi: "subvimage A f g"
shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
proof (rule disjCI)
assume "\<not> ?dist"
then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
thus "?sub" using svi unfolding subvimage_def by auto
qed
lemma setsum_image_split:
assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
(is "?lhs = ?rhs")
proof -
have "f ` A =
snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
(is "_ = snd ` ?SIGMA")
unfolding image_split_eq_Sigma[symmetric]
by (simp add: image_compose[symmetric] comp_def)
moreover
have snd_inj: "inj_on snd ?SIGMA"
unfolding image_split_eq_Sigma[symmetric]
by (auto intro!: inj_onI subvimageD[OF svi])
ultimately
have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
by (auto simp: setsum_reindex intro: setsum_cong)
also have "... = ?rhs"
using subvimage_finite[OF svi fin] fin
apply (subst setsum_Sigma[symmetric])
by (auto intro!: finite_subset[of _ "f`A"])
finally show ?thesis .
qed
lemma (in information_space) entropy_partition:
assumes sf: "simple_function M X" "simple_function M P"
assumes svi: "subvimage (space M) X P"
shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
proof -
let "?XP x p" = "joint_distribution X P {(x, p)}"
let "?X x" = "distribution X {x}"
let "?P p" = "distribution P {p}"
note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
note finite_distribution_order[OF fX fP, simp]
have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
(\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
proof (subst setsum_image_split[OF svi],
safe intro!: setsum_mono_zero_cong_left imageI)
show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
using sf unfolding simple_function_def by auto
next
fix p x assume in_space: "p \<in> space M" "x \<in> space M"
assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
show "x \<in> P -` {P p}" by auto
next
fix p x assume in_space: "p \<in> space M" "x \<in> space M"
assume "P x = P p"
from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
by auto
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
by auto
thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
by (auto simp: distribution_def)
qed
moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
by (auto simp add: log_simps zero_less_mult_iff field_simps)
ultimately show ?thesis
unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
using setsum_joint_distribution_singleton[OF fX fP]
by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
qed
corollary (in information_space) entropy_data_processing:
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
proof -
note X
moreover have fX: "simple_function M (f \<circ> X)" using X by auto
moreover have "subvimage (space M) X (f \<circ> X)" by auto
ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
then show "\<H>(f \<circ> X) \<le> \<H>(X)"
by (auto intro: conditional_entropy_positive[OF X fX])
qed
corollary (in information_space) entropy_of_inj:
assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
shows "\<H>(f \<circ> X) = \<H>(X)"
proof (rule antisym)
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
next
have sf: "simple_function M (f \<circ> X)"
using X by auto
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
also have "... \<le> \<H>(f \<circ> X)"
using entropy_data_processing[OF sf] .
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
qed
end