--- a/src/HOL/Probability/Information.thy Sat May 01 16:13:24 2010 -0700
+++ b/src/HOL/Probability/Information.thy Mon May 03 10:28:19 2010 -0700
@@ -1,169 +1,264 @@
theory Information
-imports Probability_Space Product_Measure
+imports Probability_Space Product_Measure "../Multivariate_Analysis/Convex"
begin
-lemma pos_neg_part_abs:
- fixes f :: "'a \<Rightarrow> real"
- shows "pos_part f x + neg_part f x = \<bar>f x\<bar>"
-unfolding real_abs_def pos_part_def neg_part_def by auto
+section "Convex theory"
-lemma pos_part_abs:
- fixes f :: "'a \<Rightarrow> real"
- shows "pos_part (\<lambda> x. \<bar>f x\<bar>) y = \<bar>f y\<bar>"
-unfolding pos_part_def real_abs_def by auto
-
-lemma neg_part_abs:
- fixes f :: "'a \<Rightarrow> real"
- shows "neg_part (\<lambda> x. \<bar>f x\<bar>) y = 0"
-unfolding neg_part_def real_abs_def by auto
+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 (in measure_space) int_abs:
- assumes "integrable f"
- shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
-using assms
+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 -
- from assms obtain p q where pq: "p \<in> nnfis (pos_part f)" "q \<in> nnfis (neg_part f)"
- unfolding integrable_def by auto
- hence "p + q \<in> nnfis (\<lambda> x. pos_part f x + neg_part f x)"
- using nnfis_add by auto
- hence "p + q \<in> nnfis (\<lambda> x. \<bar>f x\<bar>)" using pos_neg_part_abs[of f] by simp
- thus ?thesis unfolding integrable_def
- using ext[OF pos_part_abs[of f], of "\<lambda> y. y"]
- ext[OF neg_part_abs[of f], of "\<lambda> y. y"]
- using nnfis_0 by auto
+ 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 (in measure_space) measure_mono:
- assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M"
- shows "measure M a \<le> measure M b"
+section "Information theory"
+
+lemma (in finite_prob_space) sum_over_space_distrib:
+ "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst measure_finitely_additive'')
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+
+locale finite_information_space = finite_prob_space +
+ fixes b :: real assumes b_gt_1: "1 < b"
+
+definition
+ "KL_divergence b M X Y =
+ measure_space.integral (M\<lparr>measure := X\<rparr>)
+ (\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := Y\<rparr> ) X) x))"
+
+lemma (in finite_prob_space) distribution_mono:
+ assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+ shows "distribution X x \<le> distribution Y y"
+ unfolding distribution_def
+ using assms by (auto simp: sets_eq_Pow intro!: measure_mono)
+
+lemma (in prob_space) distribution_remove_const:
+ shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
+ and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
+ and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
+ and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
+ and "distribution (\<lambda>x. ()) {()} = 1"
+ unfolding prob_space[symmetric]
+ by (auto intro!: arg_cong[where f=prob] simp: distribution_def)
+
+
+context finite_information_space
+begin
+
+lemma distribution_mono_gt_0:
+ assumes gt_0: "0 < distribution X x"
+ assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+ shows "0 < distribution Y y"
+ by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
+
+lemma
+ assumes "0 \<le> A" and pos: "0 < A \<Longrightarrow> 0 < B" "0 < A \<Longrightarrow> 0 < C"
+ shows mult_log_mult: "A * log b (B * C) = A * log b B + A * log b C" (is "?mult")
+ and mult_log_divide: "A * log b (B / C) = A * log b B - A * log b C" (is "?div")
proof -
- have "b = a \<union> (b - a)" using assms by auto
- moreover have "{} = a \<inter> (b - a)" by auto
- ultimately have "measure M b = measure M a + measure M (b - a)"
- using measure_additive[of a "b - a"] local.Diff[of b a] assms by auto
- moreover have "measure M (b - a) \<ge> 0" using positive assms by auto
- ultimately show "measure M a \<le> measure M b" by auto
+ have "?mult \<and> ?div"
+proof (cases "A = 0")
+ case False
+ hence "0 < A" using `0 \<le> A` by auto
+ with pos[OF this] show "?mult \<and> ?div" using b_gt_1
+ by (auto simp: log_divide log_mult field_simps)
+qed simp
+ thus ?mult and ?div by auto
qed
-lemma (in measure_space) integral_0:
- fixes f :: "'a \<Rightarrow> real"
- assumes "integrable f" "integral f = 0" "nonneg f" and borel: "f \<in> borel_measurable M"
- shows "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0"
-proof -
- have "{x. f x \<noteq> 0} = {x. \<bar>f x\<bar> > 0}" by auto
- moreover
- { fix y assume "y \<in> {x. \<bar> f x \<bar> > 0}"
- hence "\<bar> f y \<bar> > 0" by auto
- hence "\<exists> n. \<bar>f y\<bar> \<ge> inverse (real (Suc n))"
- using ex_inverse_of_nat_Suc_less[of "\<bar>f y\<bar>"] less_imp_le unfolding real_of_nat_def by auto
- hence "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
- by auto }
- moreover
- { fix y assume "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
- then obtain n where n: "y \<in> {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}" by auto
- hence "\<bar>f y\<bar> \<ge> inverse (real (Suc n))" by auto
- hence "\<bar>f y\<bar> > 0"
- using real_of_nat_Suc_gt_zero
- positive_imp_inverse_positive[of "real_of_nat (Suc n)"] by fastsimp
- hence "y \<in> {x. \<bar>f x\<bar> > 0}" by auto }
- ultimately have fneq0_UN: "{x. f x \<noteq> 0} = (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
- by blast
- { fix n
- have int_one: "integrable (\<lambda> x. \<bar>f x\<bar> ^ 1)" using int_abs assms by auto
- have "measure M (f -` {inverse (real (Suc n))..} \<inter> space M)
- \<le> integral (\<lambda> x. \<bar>f x\<bar> ^ 1) / (inverse (real (Suc n)) ^ 1)"
- using markov_ineq[OF `integrable f` _ int_one] real_of_nat_Suc_gt_zero by auto
- hence le0: "measure M (f -` {inverse (real (Suc n))..} \<inter> space M) \<le> 0"
- using assms unfolding nonneg_def by auto
- have "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
- apply (subst Int_commute) unfolding Int_def
- using borel[unfolded borel_measurable_ge_iff] by simp
- hence m0: "measure M ({x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0 \<and>
- {x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
- using positive le0 unfolding atLeast_def by fastsimp }
- moreover hence "range (\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) \<subseteq> sets M"
- by auto
- moreover
- { fix n
- have "inverse (real (Suc n)) \<ge> inverse (real (Suc (Suc n)))"
- using less_imp_inverse_less real_of_nat_Suc_gt_zero[of n] by fastsimp
- hence "\<And> x. f x \<ge> inverse (real (Suc n)) \<Longrightarrow> f x \<ge> inverse (real (Suc (Suc n)))" by (rule order_trans)
- hence "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M
- \<subseteq> {x. f x \<ge> inverse (real (Suc (Suc n)))} \<inter> space M" by auto }
- ultimately have "(\<lambda> x. 0) ----> measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M)"
- using monotone_convergence[of "\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M"]
- unfolding o_def by (simp del: of_nat_Suc)
- hence "measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0"
- using LIMSEQ_const[of 0] LIMSEQ_unique by simp
- hence "measure M ((\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}) \<inter> space M) = 0"
- using assms unfolding nonneg_def by auto
- thus "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0" using fneq0_UN by simp
+lemma split_pairs:
+ shows
+ "((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
+
+ML {*
+
+ (* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
+ where @{term W} is a joint distribution of @{term X}, @{term Y}, and @{term Z}. *)
+
+ val mult_log_intros = [@{thm mult_log_divide}, @{thm mult_log_mult}]
+ val intros = [@{thm divide_pos_pos}, @{thm mult_pos_pos}, @{thm positive_distribution}]
+
+ val distribution_gt_0_tac = (rtac @{thm distribution_mono_gt_0}
+ THEN' assume_tac
+ THEN' clarsimp_tac (clasimpset_of @{context} addsimps2 @{thms split_pairs}))
+
+ val distr_mult_log_eq_tac = REPEAT_ALL_NEW (CHANGED o TRY o
+ (resolve_tac (mult_log_intros @ intros)
+ ORELSE' distribution_gt_0_tac
+ ORELSE' clarsimp_tac (clasimpset_of @{context})))
+
+ fun instanciate_term thy redex intro =
+ let
+ val intro_concl = Thm.concl_of intro
+
+ val lhs = intro_concl |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst
+
+ val m = SOME (Pattern.match thy (lhs, redex) (Vartab.empty, Vartab.empty))
+ handle Pattern.MATCH => NONE
+
+ in
+ Option.map (fn m => Envir.subst_term m intro_concl) m
+ end
+
+ fun mult_log_simproc simpset redex =
+ let
+ val ctxt = Simplifier.the_context simpset
+ val thy = ProofContext.theory_of ctxt
+ fun prove (SOME thm) = (SOME
+ (Goal.prove ctxt [] [] thm (K (distr_mult_log_eq_tac 1))
+ |> mk_meta_eq)
+ handle THM _ => NONE)
+ | prove NONE = NONE
+ in
+ get_first (instanciate_term thy (term_of redex) #> prove) mult_log_intros
+ end
+*}
+
+simproc_setup mult_log ("distribution X x * log b (A * B)" |
+ "distribution X x * log b (A / B)") = {* K mult_log_simproc *}
+
+end
+
+lemma KL_divergence_eq_finite:
+ assumes u: "finite_measure_space (M\<lparr>measure := u\<rparr>)"
+ assumes v: "finite_measure_space (M\<lparr>measure := v\<rparr>)"
+ assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
+ shows "KL_divergence b M u v = (\<Sum>x\<in>space M. u {x} * log b (u {x} / v {x}))" (is "_ = ?sum")
+proof (simp add: KL_divergence_def, subst finite_measure_space.integral_finite_singleton, simp_all add: u)
+ have ms_u: "measure_space (M\<lparr>measure := u\<rparr>)"
+ using u unfolding finite_measure_space_def by simp
+
+ show "(\<Sum>x \<in> space M. log b (measure_space.RN_deriv (M\<lparr>measure := v\<rparr>) u x) * u {x}) = ?sum"
+ apply (rule setsum_cong[OF refl])
+ apply simp
+ apply (safe intro!: arg_cong[where f="log b"] )
+ apply (subst finite_measure_space.RN_deriv_finite_singleton)
+ using assms ms_u by auto
qed
-definition
- "KL_divergence b M u v =
- measure_space.integral (M\<lparr>measure := u\<rparr>)
- (\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := v\<rparr> ) u) x))"
-
-lemma (in finite_prob_space) finite_measure_space:
- shows "finite_measure_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
- (is "finite_measure_space ?S")
-proof (rule finite_Pow_additivity_sufficient, simp_all)
- show "finite (X ` space M)" using finite_space by simp
-
- show "positive ?S (distribution X)" unfolding distribution_def
- unfolding positive_def using positive'[unfolded positive_def] sets_eq_Pow by auto
+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
- show "additive ?S (distribution X)" unfolding additive_def distribution_def
- proof (simp, safe)
- fix x y
- have x: "(X -` x) \<inter> space M \<in> sets M"
- and y: "(X -` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto
- assume "x \<inter> y = {}"
- from additive[unfolded additive_def, rule_format, OF x y] this
- have "prob (((X -` x) \<union> (X -` y)) \<inter> space M) =
- prob ((X -` x) \<inter> space M) + prob ((X -` y) \<inter> space M)"
- apply (subst Int_Un_distrib2)
- by auto
- thus "prob ((X -` x \<union> X -` y) \<inter> space M) = prob (X -` x \<inter> space M) + prob (X -` y \<inter> space M)"
- by 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 (in finite_prob_space) finite_prob_space:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
- (is "finite_prob_space ?S")
- unfolding finite_prob_space_def prob_space_def prob_space_axioms_def
-proof safe
- show "finite_measure_space ?S" by (rule finite_measure_space)
- thus "measure_space ?S" by (simp add: finite_measure_space_def)
+lemma KL_divergence_positive_finite:
+ assumes u: "finite_prob_space (M\<lparr>measure := u\<rparr>)"
+ assumes v: "finite_prob_space (M\<lparr>measure := v\<rparr>)"
+ assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
+ and "1 < b"
+ shows "0 \<le> KL_divergence b M u v"
+proof -
+ interpret u: finite_prob_space "M\<lparr>measure := u\<rparr>" using u .
+ interpret v: finite_prob_space "M\<lparr>measure := v\<rparr>" using v .
- have "X -` X ` space M \<inter> space M = space M" by auto
- thus "measure ?S (space ?S) = 1"
- by (simp add: distribution_def prob_space)
-qed
+ have *: "space M \<noteq> {}" using u.not_empty by simp
-lemma (in finite_prob_space) finite_measure_space_image_prod:
- "finite_measure_space \<lparr>space = X ` space M \<times> Y ` space M,
- sets = Pow (X ` space M \<times> Y ` space M), measure_space.measure = distribution (\<lambda>x. (X x, Y x))\<rparr>"
- (is "finite_measure_space ?Z")
-proof (rule finite_Pow_additivity_sufficient, simp_all)
- show "finite (X ` space M \<times> Y ` space M)" using finite_space by simp
+ have "- (KL_divergence b M u v) \<le> log b (\<Sum>x\<in>space M. v {x})"
+ proof (subst KL_divergence_eq_finite, safe intro!: log_setsum_divide *)
+ show "finite_measure_space (M\<lparr>measure := u\<rparr>)"
+ "finite_measure_space (M\<lparr>measure := v\<rparr>)"
+ using u v unfolding finite_prob_space_eq by simp_all
- let ?d = "distribution (\<lambda>x. (X x, Y x))"
+ show "finite (space M)" using u.finite_space by simp
+ show "1 < b" by fact
+ show "(\<Sum>x\<in>space M. u {x}) = 1" using u.sum_over_space_eq_1 by simp
- show "positive ?Z ?d"
- using sets_eq_Pow by (auto simp: positive_def distribution_def intro!: positive)
+ fix x assume x: "x \<in> space M"
+ thus pos: "0 \<le> u {x}" "0 \<le> v {x}"
+ using u.positive u.sets_eq_Pow v.positive v.sets_eq_Pow by simp_all
- show "additive ?Z ?d" unfolding additive_def
- proof safe
- fix x y assume "x \<in> sets ?Z" and "y \<in> sets ?Z"
- assume "x \<inter> y = {}"
- thus "?d (x \<union> y) = ?d x + ?d y"
- apply (simp add: distribution_def)
- apply (subst measure_additive[unfolded sets_eq_Pow, simplified])
- by (auto simp: Un_Int_distrib Un_Int_distrib2 intro!: arg_cong[where f=prob])
+ { assume "v {x} = 0" from u_0[OF x this] show "u {x} = 0" . }
+ { assume "0 < u {x}"
+ hence "v {x} \<noteq> 0" using u_0[OF x] by auto
+ with pos show "0 < v {x}" by simp }
qed
+ thus "0 \<le> KL_divergence b M u v" using v.sum_over_space_eq_1 by simp
qed
definition (in prob_space)
@@ -174,346 +269,142 @@
in
KL_divergence b prod_space (joint_distribution X Y) (measure prod_space)"
-abbreviation (in finite_prob_space)
- finite_mutual_information ("\<I>\<^bsub>_\<^esub>'(_ ; _')") where
- "\<I>\<^bsub>b\<^esub>(X ; Y) \<equiv> mutual_information b
+abbreviation (in finite_information_space)
+ finite_mutual_information ("\<I>'(_ ; _')") where
+ "\<I>(X ; Y) \<equiv> mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
-abbreviation (in finite_prob_space)
- finite_mutual_information_2 :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'd) \<Rightarrow> real" ("\<I>'(_ ; _')") where
- "\<I>(X ; Y) \<equiv> \<I>\<^bsub>2\<^esub>(X ; Y)"
+lemma (in finite_measure_space) measure_spaceI: "measure_space M"
+ by unfold_locales
-lemma (in prob_space) mutual_information_cong:
- assumes [simp]: "space S1 = space S3" "sets S1 = sets S3"
- "space S2 = space S4" "sets S2 = sets S4"
- shows "mutual_information b S1 S2 X Y = mutual_information b S3 S4 X Y"
- unfolding mutual_information_def by simp
+lemma prod_measure_times_finite:
+ assumes fms: "finite_measure_space M" "finite_measure_space M'" and a: "a \<in> space M \<times> space M'"
+ shows "prod_measure M M' {a} = measure M {fst a} * measure M' {snd a}"
+proof (cases a)
+ case (Pair b c)
+ hence a_eq: "{a} = {b} \<times> {c}" by simp
-lemma (in prob_space) joint_distribution:
- "joint_distribution X Y = distribution (\<lambda>x. (X x, Y x))"
- unfolding joint_distribution_def_raw distribution_def_raw ..
+ with fms[THEN finite_measure_space.measure_spaceI]
+ fms[THEN finite_measure_space.sets_eq_Pow] a Pair
+ show ?thesis unfolding a_eq
+ by (subst prod_measure_times) simp_all
+qed
-lemma (in finite_prob_space) finite_mutual_information_reduce:
- "\<I>\<^bsub>b\<^esub>(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})))"
- (is "_ = setsum ?log ?prod")
- unfolding Let_def mutual_information_def KL_divergence_def
-proof (subst finite_measure_space.integral_finite_singleton, simp_all add: joint_distribution)
- let ?X = "\<lparr>space = X ` space M, sets = Pow (X ` space M), measure_space.measure = distribution X\<rparr>"
- let ?Y = "\<lparr>space = Y ` space M, sets = Pow (Y ` space M), measure_space.measure = distribution Y\<rparr>"
- let ?P = "prod_measure_space ?X ?Y"
+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
- interpret X: finite_measure_space "?X" by (rule finite_measure_space)
- moreover interpret Y: finite_measure_space "?Y" by (rule finite_measure_space)
- ultimately have ms_X: "measure_space ?X" and ms_Y: "measure_space ?Y" by unfold_locales
-
- interpret P: finite_measure_space "?P" by (rule finite_measure_space_finite_prod_measure) (fact+)
-
- let ?P' = "measure_update (\<lambda>_. distribution (\<lambda>x. (X x, Y x))) ?P"
- from finite_measure_space_image_prod[of X Y]
- sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
- show "finite_measure_space ?P'"
- by (simp add: X.sets_eq_Pow Y.sets_eq_Pow joint_distribution finite_measure_space_def prod_measure_space_def)
+lemma (in finite_information_space)
+ assumes MX: "finite_prob_space \<lparr> space = space MX, sets = sets MX, measure = distribution X\<rparr>"
+ (is "finite_prob_space ?MX")
+ assumes MY: "finite_prob_space \<lparr> space = space MY, sets = sets MY, measure = distribution Y\<rparr>"
+ (is "finite_prob_space ?MY")
+ and X_space: "X ` space M \<subseteq> space MX" and Y_space: "Y ` space M \<subseteq> space MY"
+ shows mutual_information_eq_generic:
+ "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 "?equality")
+ and mutual_information_positive_generic:
+ "0 \<le> mutual_information b MX MY X Y" (is "?positive")
+proof -
+ let ?P = "prod_measure_space ?MX ?MY"
+ let ?measure = "joint_distribution X Y"
+ let ?P' = "measure_update (\<lambda>_. ?measure) ?P"
- show "(\<Sum>x \<in> space ?P. log b (measure_space.RN_deriv ?P (distribution (\<lambda>x. (X x, Y x))) x) * distribution (\<lambda>x. (X x, Y x)) {x})
- = setsum ?log ?prod"
- proof (rule setsum_cong)
- show "space ?P = ?prod" unfolding prod_measure_space_def by simp
- next
- fix x assume x: "x \<in> X ` space M \<times> Y ` space M"
- then obtain d e where x_Pair: "x = (d, e)"
- and d: "d \<in> X ` space M"
- and e: "e \<in> Y ` space M" by auto
-
- { fix x assume m_0: "measure ?P {x} = 0"
- have "distribution (\<lambda>x. (X x, Y x)) {x} = 0"
- proof (cases x)
- case (Pair a b)
- hence "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = (X -` {a} \<inter> space M) \<inter> (Y -` {b} \<inter> space M)"
- and x_prod: "{x} = {a} \<times> {b}" by auto
+ interpret X: finite_prob_space "?MX" using MX .
+ moreover interpret Y: finite_prob_space "?MY" using MY .
+ ultimately have ms_X: "measure_space ?MX"
+ and ms_Y: "measure_space ?MY" by unfold_locales
- let ?PROD = "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M"
+ have fms_P: "finite_measure_space ?P"
+ by (rule finite_measure_space_finite_prod_measure) fact+
+
+ have fms_P': "finite_measure_space ?P'"
+ using finite_product_measure_space[of "space MX" "space MY"]
+ X.finite_space Y.finite_space sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
+ X.sets_eq_Pow Y.sets_eq_Pow
+ by (simp add: prod_measure_space_def)
- show ?thesis
- proof (cases "{a} \<subseteq> X ` space M \<and> {b} \<subseteq> Y ` space M")
- case False
- hence "?PROD = {}"
- unfolding Pair by auto
- thus ?thesis by (auto simp: distribution_def)
- next
- have [intro]: "prob ?PROD \<le> 0 \<Longrightarrow> prob ?PROD = 0"
- using sets_eq_Pow by (auto intro!: positive real_le_antisym[of _ 0])
+ { fix x assume "x \<in> space ?P"
+ hence x_in_MX: "{fst x} \<in> sets MX" using X.sets_eq_Pow
+ by (auto simp: prod_measure_space_def)
+
+ assume "measure ?P {x} = 0"
+ with prod_measure_times[OF ms_X ms_Y, of "{fst x}" "{snd x}"] x_in_MX
+ have "distribution X {fst x} = 0 \<or> distribution Y {snd x} = 0"
+ by (simp add: prod_measure_space_def)
+
+ hence "joint_distribution X Y {x} = 0"
+ by (cases x) (auto simp: distribution_order) }
+ note measure_0 = this
- case True
- with prod_measure_times[OF ms_X ms_Y, simplified, of "{a}" "{b}"]
- have "prob (X -` {a} \<inter> space M) = 0 \<or> prob (Y -` {b} \<inter> space M) = 0" (is "?X_0 \<or> ?Y_0") using m_0
- by (simp add: prod_measure_space_def distribution_def Pair)
- thus ?thesis
- proof (rule disjE)
- assume ?X_0
- have "prob ?PROD \<le> prob (X -` {a} \<inter> space M)"
- using sets_eq_Pow Pair by (auto intro!: measure_mono)
- thus ?thesis using `?X_0` by (auto simp: distribution_def)
- next
- assume ?Y_0
- have "prob ?PROD \<le> prob (Y -` {b} \<inter> space M)"
- using sets_eq_Pow Pair by (auto intro!: measure_mono)
- thus ?thesis using `?Y_0` by (auto simp: distribution_def)
- qed
- qed
- qed }
- note measure_zero_joint_distribution = this
+ show ?equality
+ unfolding Let_def mutual_information_def using fms_P fms_P' measure_0 MX MY
+ by (subst KL_divergence_eq_finite)
+ (simp_all add: prod_measure_space_def prod_measure_times_finite
+ finite_prob_space_eq setsum_cartesian_product')
- show "log b (measure_space.RN_deriv ?P (distribution (\<lambda>x. (X x, Y x))) x) * distribution (\<lambda>x. (X x, Y x)) {x} = ?log x"
- apply (cases "distribution (\<lambda>x. (X x, Y x)) {x} \<noteq> 0")
- apply (subst P.RN_deriv_finite_singleton)
- proof (simp_all add: x_Pair)
- from `finite_measure_space ?P'` show "measure_space ?P'" by (simp add: finite_measure_space_def)
- next
- fix x assume m_0: "measure ?P {x} = 0" thus "distribution (\<lambda>x. (X x, Y x)) {x} = 0" by fact
- next
- show "(d,e) \<in> space ?P" unfolding prod_measure_space_def using x x_Pair by simp
- next
- assume jd_0: "distribution (\<lambda>x. (X x, Y x)) {(d, e)} \<noteq> 0"
- show "measure ?P {(d,e)} \<noteq> 0"
- proof
- assume "measure ?P {(d,e)} = 0"
- from measure_zero_joint_distribution[OF this] jd_0
- show False by simp
- qed
- next
- assume jd_0: "distribution (\<lambda>x. (X x, Y x)) {(d, e)} \<noteq> 0"
- with prod_measure_times[OF ms_X ms_Y, simplified, of "{d}" "{e}"] d
- show "log b (distribution (\<lambda>x. (X x, Y x)) {(d, e)} / measure ?P {(d, e)}) =
- log b (distribution (\<lambda>x. (X x, Y x)) {(d, e)} / (distribution X {d} * distribution Y {e}))"
- by (auto intro!: arg_cong[where f="log b"] simp: prod_measure_space_def)
- qed
+ show ?positive
+ unfolding Let_def mutual_information_def using measure_0 b_gt_1
+ proof (safe intro!: KL_divergence_positive_finite, simp_all)
+ from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space
+ have "measure ?P (space ?P) = 1"
+ by (simp add: prod_measure_space_def, subst prod_measure_times, simp_all)
+ with fms_P show "finite_prob_space ?P"
+ by (simp add: finite_prob_space_eq)
+
+ from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space Y.not_empty X_space Y_space
+ have "measure ?P' (space ?P') = 1" unfolding prob_space[symmetric]
+ by (auto simp add: prod_measure_space_def distribution_def vimage_Times comp_def
+ intro!: arg_cong[where f=prob])
+ with fms_P' show "finite_prob_space ?P'"
+ by (simp add: finite_prob_space_eq)
qed
qed
-lemma (in finite_prob_space) distribution_log_split:
- assumes "1 < b"
- shows
- "distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(X x, z)} /
- (distribution X {X x} * distribution Z {z})) =
- distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(X x, z)} /
- distribution Z {z}) -
- distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution X {X x})"
- (is "?lhs = ?rhs")
-proof (cases "distribution (\<lambda>x. (X x, Z x)) {(X x, z)} = 0")
- case True thus ?thesis by simp
-next
- case False
-
- let ?dZ = "distribution Z"
- let ?dX = "distribution X"
- let ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
-
- have dist_nneg: "\<And>x X. 0 \<le> distribution X x"
- unfolding distribution_def using sets_eq_Pow by (auto intro: positive)
-
- have "?lhs = ?dXZ {(X x, z)} * (log b (?dXZ {(X x, z)} / ?dZ {z}) - log b (?dX {X x}))"
- proof -
- have pos_dXZ: "0 < ?dXZ {(X x, z)}"
- using False dist_nneg[of "\<lambda>x. (X x, Z x)" "{(X x, z)}"] by auto
- moreover
- have "((\<lambda>x. (X x, Z x)) -` {(X x, z)}) \<inter> space M \<subseteq> (X -` {X x}) \<inter> space M" by auto
- hence "?dXZ {(X x, z)} \<le> ?dX {X x}"
- unfolding distribution_def
- by (rule measure_mono) (simp_all add: sets_eq_Pow)
- with pos_dXZ have "0 < ?dX {X x}" by (rule less_le_trans)
- moreover
- have "((\<lambda>x. (X x, Z x)) -` {(X x, z)}) \<inter> space M \<subseteq> (Z -` {z}) \<inter> space M" by auto
- hence "?dXZ {(X x, z)} \<le> ?dZ {z}"
- unfolding distribution_def
- by (rule measure_mono) (simp_all add: sets_eq_Pow)
- with pos_dXZ have "0 < ?dZ {z}" by (rule less_le_trans)
- moreover have "0 < b" by (rule less_trans[OF _ `1 < b`]) simp
- moreover have "b \<noteq> 1" by (rule ccontr) (insert `1 < b`, simp)
- ultimately show ?thesis
- using pos_dXZ
- apply (subst (2) mult_commute)
- apply (subst divide_divide_eq_left[symmetric])
- apply (subst log_divide)
- by (auto intro: divide_pos_pos)
- qed
- also have "... = ?rhs"
- by (simp add: field_simps)
- finally show ?thesis .
-qed
-
-lemma (in finite_prob_space) finite_mutual_information_reduce_prod:
- "mutual_information b
- \<lparr> space = X ` space M, sets = Pow (X ` space M) \<rparr>
- \<lparr> space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M) \<rparr>
- X (\<lambda>x. (Y x,Z x)) =
- (\<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)} /
- (distribution X {x} * distribution (\<lambda>x. (Y x,Z x)) {(y,z)})))" (is "_ = setsum ?log ?space")
- unfolding Let_def mutual_information_def KL_divergence_def using finite_space
-proof (subst finite_measure_space.integral_finite_singleton,
- simp_all add: prod_measure_space_def sigma_prod_sets_finite joint_distribution)
- let ?sets = "Pow (X ` space M \<times> Y ` space M \<times> Z ` space M)"
- and ?measure = "distribution (\<lambda>x. (X x, Y x, Z x))"
- let ?P = "\<lparr> space = ?space, sets = ?sets, measure = ?measure\<rparr>"
-
- show "finite_measure_space ?P"
- proof (rule finite_Pow_additivity_sufficient, simp_all)
- show "finite ?space" using finite_space by auto
-
- show "positive ?P ?measure"
- using sets_eq_Pow by (auto simp: positive_def distribution_def intro!: positive)
-
- show "additive ?P ?measure"
- proof (simp add: additive_def distribution_def, safe)
- fix x y assume "x \<subseteq> ?space" and "y \<subseteq> ?space"
- assume "x \<inter> y = {}"
- thus "prob (((\<lambda>x. (X x, Y x, Z x)) -` x \<union> (\<lambda>x. (X x, Y x, Z x)) -` y) \<inter> space M) =
- prob ((\<lambda>x. (X x, Y x, Z x)) -` x \<inter> space M) + prob ((\<lambda>x. (X x, Y x, Z x)) -` y \<inter> space M)"
- apply (subst measure_additive[unfolded sets_eq_Pow, simplified])
- by (auto simp: Un_Int_distrib Un_Int_distrib2 intro!: arg_cong[where f=prob])
- qed
- qed
+lemma (in finite_information_space) mutual_information_eq:
+ "\<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})))"
+ by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
- let ?X = "\<lparr>space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
- and ?YZ = "\<lparr>space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M), measure = distribution (\<lambda>x. (Y x, Z x))\<rparr>"
- let ?u = "prod_measure ?X ?YZ"
-
- from finite_measure_space[of X] finite_measure_space_image_prod[of Y Z]
- have ms_X: "measure_space ?X" and ms_YZ: "measure_space ?YZ"
- by (simp_all add: finite_measure_space_def)
-
- show "(\<Sum>x \<in> ?space. log b (measure_space.RN_deriv \<lparr>space=?space, sets=?sets, measure=?u\<rparr>
- (distribution (\<lambda>x. (X x, Y x, Z x))) x) * distribution (\<lambda>x. (X x, Y x, Z x)) {x})
- = setsum ?log ?space"
- proof (rule setsum_cong)
- fix x assume x: "x \<in> ?space"
- then obtain d e f where x_Pair: "x = (d, e, f)"
- and d: "d \<in> X ` space M"
- and e: "e \<in> Y ` space M"
- and f: "f \<in> Z ` space M" by auto
-
- { fix x assume m_0: "?u {x} = 0"
-
- let ?PROD = "(\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M"
- obtain a b c where Pair: "x = (a, b, c)" by (cases x)
- hence "?PROD = (X -` {a} \<inter> space M) \<inter> ((\<lambda>x. (Y x, Z x)) -` {(b, c)} \<inter> space M)"
- and x_prod: "{x} = {a} \<times> {(b, c)}" by auto
-
- have "distribution (\<lambda>x. (X x, Y x, Z x)) {x} = 0"
- proof (cases "{a} \<subseteq> X ` space M")
- case False
- hence "?PROD = {}"
- unfolding Pair by auto
- thus ?thesis by (auto simp: distribution_def)
- next
- have [intro]: "prob ?PROD \<le> 0 \<Longrightarrow> prob ?PROD = 0"
- using sets_eq_Pow by (auto intro!: positive real_le_antisym[of _ 0])
-
- case True
- with prod_measure_times[OF ms_X ms_YZ, simplified, of "{a}" "{(b,c)}"]
- have "prob (X -` {a} \<inter> space M) = 0 \<or> prob ((\<lambda>x. (Y x, Z x)) -` {(b, c)} \<inter> space M) = 0"
- (is "prob ?X = 0 \<or> prob ?Y = 0") using m_0
- by (simp add: prod_measure_space_def distribution_def Pair)
- thus ?thesis
- proof (rule disjE)
- assume "prob ?X = 0"
- have "prob ?PROD \<le> prob ?X"
- using sets_eq_Pow Pair by (auto intro!: measure_mono)
- thus ?thesis using `prob ?X = 0` by (auto simp: distribution_def)
- next
- assume "prob ?Y = 0"
- have "prob ?PROD \<le> prob ?Y"
- using sets_eq_Pow Pair by (auto intro!: measure_mono)
- thus ?thesis using `prob ?Y = 0` by (auto simp: distribution_def)
- qed
- qed }
- note measure_zero_joint_distribution = this
-
- from x_Pair d e f finite_space
- show "log b (measure_space.RN_deriv \<lparr>space=?space, sets=?sets, measure=?u\<rparr>
- (distribution (\<lambda>x. (X x, Y x, Z x))) x) * distribution (\<lambda>x. (X x, Y x, Z x)) {x} = ?log x"
- apply (cases "distribution (\<lambda>x. (X x, Y x, Z x)) {x} \<noteq> 0")
- apply (subst finite_measure_space.RN_deriv_finite_singleton)
- proof simp_all
- show "measure_space ?P" using `finite_measure_space ?P` by (simp add: finite_measure_space_def)
-
- from finite_measure_space_finite_prod_measure[OF finite_measure_space[of X]
- finite_measure_space_image_prod[of Y Z]] finite_space
- show "finite_measure_space \<lparr>space=?space, sets=?sets, measure=?u\<rparr>"
- by (simp add: prod_measure_space_def sigma_prod_sets_finite)
- next
- fix x assume "?u {x} = 0" thus "distribution (\<lambda>x. (X x, Y x, Z x)) {x} = 0" by fact
- next
- assume jd_0: "distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} \<noteq> 0"
- show "?u {(d,e,f)} \<noteq> 0"
- proof
- assume "?u {(d, e, f)} = 0"
- from measure_zero_joint_distribution[OF this] jd_0
- show False by simp
- qed
- next
- assume jd_0: "distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} \<noteq> 0"
- with prod_measure_times[OF ms_X ms_YZ, simplified, of "{d}" "{(e,f)}"] d
- show "log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} / ?u {(d, e, f)}) =
- log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} / (distribution X {d} * distribution (\<lambda>x. (Y x, Z x)) {(e,f)}))"
- by (auto intro!: arg_cong[where f="log b"] simp: prod_measure_space_def)
- qed
- qed simp
-qed
+lemma (in finite_information_space) mutual_information_positive: "0 \<le> \<I>(X;Y)"
+ by (subst mutual_information_positive_generic) (simp_all add: finite_prob_space_of_images)
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
-abbreviation (in finite_prob_space)
- finite_entropy ("\<H>\<^bsub>_\<^esub>'(_')") where
- "\<H>\<^bsub>b\<^esub>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
-
-abbreviation (in finite_prob_space)
- finite_entropy_2 ("\<H>'(_')") where
- "\<H>(X) \<equiv> \<H>\<^bsub>2\<^esub>(X)"
+abbreviation (in finite_information_space)
+ finite_entropy ("\<H>'(_')") where
+ "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
-lemma (in finite_prob_space) finite_entropy_reduce:
- assumes "1 < b"
- shows "\<H>\<^bsub>b\<^esub>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
+lemma (in finite_information_space) joint_distribution_remove[simp]:
+ "joint_distribution X X {(x, x)} = distribution X {x}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=prob])
+
+lemma (in finite_information_space) entropy_eq:
+ "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
proof -
- have fin: "finite (X ` space M)" using finite_space by simp
-
- have If_mult_distr: "\<And>A B C D. If A B C * D = If A (B * D) (C * D)" by auto
-
+ { fix f
{ fix x y
have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
- hence "distribution (\<lambda>x. (X x, X x)) {(x,y)} = (if x = y then distribution X {x} else 0)"
+ hence "distribution (\<lambda>x. (X x, X x)) {(x,y)} * f x y = (if x = y then distribution X {x} * f x y else 0)"
unfolding distribution_def by auto }
- moreover
- have "\<And>x. 0 \<le> distribution X x"
- unfolding distribution_def using finite_space sets_eq_Pow by (auto intro: positive)
- hence "\<And>x. distribution X x \<noteq> 0 \<Longrightarrow> 0 < distribution X x" by (auto simp: le_less)
- ultimately
- show ?thesis using `1 < b`
- by (auto intro!: setsum_cong
- simp: log_inverse If_mult_distr setsum_cases[OF fin] inverse_eq_divide[symmetric]
- entropy_def setsum_negf[symmetric] joint_distribution finite_mutual_information_reduce
- setsum_cartesian_product[symmetric])
+ hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. joint_distribution X X {(x, y)} * f x y) =
+ (\<Sum>x \<in> X ` space M. distribution X {x} * f x x)"
+ unfolding setsum_cartesian_product' by (simp add: setsum_cases finite_space) }
+ note remove_cartesian_product = this
+
+ show ?thesis
+ unfolding entropy_def mutual_information_eq setsum_negf[symmetric] remove_cartesian_product
+ by (auto intro!: setsum_cong)
qed
-lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
-proof (rule inj_onI, simp)
- fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
- show "x = y"
- proof (cases rule: linorder_cases)
- assume "x < y" hence "log b x < log b y"
- using log_less_cancel_iff[OF `1 < b`] pos by simp
- thus ?thesis using * by simp
- next
- assume "y < x" hence "log b y < log b x"
- using log_less_cancel_iff[OF `1 < b`] pos by simp
- thus ?thesis using * by simp
- qed simp
-qed
+lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
+ unfolding entropy_def using mutual_information_positive .
definition (in prob_space)
"conditional_mutual_information b s1 s2 s3 X Y Z \<equiv>
@@ -524,160 +415,181 @@
mutual_information b s1 prod_space X (\<lambda>x. (Y x, Z x)) -
mutual_information b s1 s3 X Z"
-abbreviation (in finite_prob_space)
- finite_conditional_mutual_information ("\<I>\<^bsub>_\<^esub>'( _ ; _ | _ ')") where
- "\<I>\<^bsub>b\<^esub>(X ; Y | Z) \<equiv> conditional_mutual_information b
+abbreviation (in finite_information_space)
+ finite_conditional_mutual_information ("\<I>'( _ ; _ | _ ')") where
+ "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr>
\<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
X Y Z"
-abbreviation (in finite_prob_space)
- finite_conditional_mutual_information_2 ("\<I>'( _ ; _ | _ ')") where
- "\<I>(X ; Y | Z) \<equiv> \<I>\<^bsub>2\<^esub>(X ; Y | Z)"
+lemma (in finite_information_space) setsum_distribution_gen:
+ assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
+ and "inj_on f (X`space M)"
+ shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
+ unfolding distribution_def assms
+ using finite_space assms
+ by (subst measure_finitely_additive'')
+ (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
+ intro!: arg_cong[where f=prob])
+
+lemma (in finite_information_space) setsum_distribution:
+ "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
+ "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
+ "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
+ "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
+ "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
+ by (auto intro!: inj_onI setsum_distribution_gen)
-lemma image_pair_eq_Sigma:
- "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
-proof (safe intro!: imageI vimageI, simp_all)
- fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
- show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A" unfolding eq[symmetric]
- using * by auto
+lemma (in finite_information_space) conditional_mutual_information_eq_sum:
+ "\<I>(X ; Y | Z) =
+ (\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` 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)}/
+ distribution (\<lambda>x. (Y x, Z x)) {(y, z)})) -
+ (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
+ distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))"
+ (is "_ = ?rhs")
+proof -
+ have setsum_product:
+ "\<And>f x. (\<Sum>v\<in>(\<lambda>x. (Y x, Z x)) ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)
+ = (\<Sum>v\<in>Y ` space M \<times> Z ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)"
+ proof (safe intro!: setsum_mono_zero_cong_left imageI)
+ fix x y z f
+ assume *: "(Y y, Z z) \<notin> (\<lambda>x. (Y x, Z x)) ` space M" and "y \<in> space M" "z \<in> space M"
+ hence "(\<lambda>x. (X x, Y x, Z x)) -` {(x, Y y, Z z)} \<inter> space M = {}"
+ proof safe
+ fix x' assume x': "x' \<in> space M" and eq: "Y x' = Y y" "Z x' = Z z"
+ have "(Y y, Z z) \<in> (\<lambda>x. (Y x, Z x)) ` space M" using eq[symmetric] x' by auto
+ thus "x' \<in> {}" using * by auto
+ qed
+ thus "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, Y y, Z z)} * f (Y y) (Z z) = 0"
+ unfolding distribution_def by simp
+ qed (simp add: finite_space)
+
+ thus ?thesis
+ unfolding conditional_mutual_information_def Let_def mutual_information_eq
+ apply (subst mutual_information_eq_generic)
+ by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
+ finite_prob_space_of_images finite_product_prob_space_of_images
+ setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
+ setsum_left_distrib[symmetric] setsum_distribution
+ cong: setsum_cong)
qed
-lemma inj_on_swap: "inj_on (\<lambda>(x,y). (y,x)) A" by (auto intro!: inj_onI)
-
-lemma (in finite_prob_space) finite_conditional_mutual_information_reduce:
- assumes "1 < b"
- shows "\<I>\<^bsub>b\<^esub>(X ; Y | Z) =
- - (\<Sum> (x, z) \<in> (X ` space M \<times> Z ` space M).
- distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))
- + (\<Sum> (x, y, z) \<in> (X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M).
+lemma (in finite_information_space) conditional_mutual_information_eq:
+ "\<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)}/
- distribution (\<lambda>x. (Y x, Z x)) {(y, z)}))" (is "_ = ?rhs")
-unfolding conditional_mutual_information_def Let_def using finite_space
-apply (simp add: prod_measure_space_def sigma_prod_sets_finite)
-apply (subst mutual_information_cong[of _ "\<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr>"
- _ "\<lparr>space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M)\<rparr>"], simp_all)
-apply (subst finite_mutual_information_reduce_prod, simp_all)
-apply (subst finite_mutual_information_reduce, simp_all)
+ (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
+ unfolding conditional_mutual_information_def Let_def mutual_information_eq
+ apply (subst mutual_information_eq_generic)
+ by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
+ finite_prob_space_of_images finite_product_prob_space_of_images
+ setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
+ setsum_left_distrib[symmetric] setsum_distribution setsum_commute[where A="Y`space M"]
+ cong: setsum_cong)
+
+lemma (in finite_information_space) conditional_mutual_information_eq_mutual_information:
+ "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
+proof -
+ have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
+
+ show ?thesis
+ unfolding conditional_mutual_information_eq mutual_information_eq
+ by (simp add: setsum_cartesian_product' distribution_remove_const)
+qed
+
+lemma (in finite_information_space) conditional_mutual_information_positive:
+ "0 \<le> \<I>(X ; Y | Z)"
proof -
let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
- let ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
- let ?dYZ = "distribution (\<lambda>x. (Y x, Z x))"
+ let ?dXZ = "joint_distribution X Z"
+ let ?dYZ = "joint_distribution Y Z"
let ?dX = "distribution X"
- let ?dY = "distribution Y"
let ?dZ = "distribution Z"
+ let ?M = "X ` space M \<times> Y ` space M \<times> Z ` space M"
+
+ have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: expand_fun_eq)
- have If_mult_distr: "\<And>A B C D. If A B C * D = If A (B * D) (C * D)" by auto
- { fix x y
- have "(\<lambda>x. (X x, Y x, Z x)) -` {(X x, y)} \<inter> space M =
- (if y \<in> (\<lambda>x. (Y x, Z x)) ` space M then (\<lambda>x. (X x, Y x, Z x)) -` {(X x, y)} \<inter> space M else {})" by auto
- hence "?dXYZ {(X x, y)} = (if y \<in> (\<lambda>x. (Y x, Z x)) ` space M then ?dXYZ {(X x, y)} else 0)"
- unfolding distribution_def by auto }
- note split_measure = this
-
- have sets: "Y ` space M \<times> Z ` space M \<inter> (\<lambda>x. (Y x, Z x)) ` space M = (\<lambda>x. (Y x, Z x)) ` space M" by auto
-
- have cong: "\<And>A B C D. \<lbrakk> A = C ; B = D \<rbrakk> \<Longrightarrow> A + B = C + D" by auto
-
- { fix A f have "setsum f A = setsum (\<lambda>(x, y). f (y, x)) ((\<lambda>(x, y). (y, x)) ` A)"
- using setsum_reindex[OF inj_on_swap, of "\<lambda>(x, y). f (y, x)" A] by (simp add: split_twice) }
- note setsum_reindex_swap = this
-
- { fix A B f assume *: "finite A" "\<forall>x\<in>A. finite (B x)"
- have "(\<Sum>x\<in>Sigma A B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) (B x))"
- unfolding setsum_Sigma[OF *] by simp }
- note setsum_Sigma = this
+ have "- (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
+ log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))
+ \<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 not_empty by simp
+ show "1 < b" using b_gt_1 .
- { fix x
- have "(\<Sum>z\<in>Z ` space M. ?dXZ {(X x, z)}) = (\<Sum>yz\<in>(\<lambda>x. (Y x, Z x)) ` space M. ?dXYZ {(X x, yz)})"
- apply (subst setsum_reindex_swap)
- apply (simp add: image_image distribution_def)
- unfolding image_pair_eq_Sigma
- apply (subst setsum_Sigma)
- using finite_space apply simp_all
- apply (rule setsum_cong[OF refl])
- apply (subst measure_finitely_additive'')
- by (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob]) }
+ fix x assume "x \<in> ?M"
+ show "0 \<le> ?dXYZ {(fst x, fst (snd x), snd (snd x))}" using positive_distribution .
+ show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+ by (auto intro!: mult_nonneg_nonneg positive_distribution simp: zero_le_divide_iff)
- thus "(\<Sum>(x, y, z)\<in>X ` space M \<times> Y ` space M \<times> Z ` space M.
- ?dXYZ {(x, y, z)} * log b (?dXYZ {(x, y, z)} / (?dX {x} * ?dYZ {(y, z)}))) -
- (\<Sum>(x, y)\<in>X ` space M \<times> Z ` space M.
- ?dXZ {(x, y)} * log b (?dXZ {(x, y)} / (?dX {x} * ?dZ {y}))) =
- - (\<Sum> (x, z) \<in> (X ` space M \<times> Z ` space M).
- ?dXZ {(x,z)} * log b (?dXZ {(x,z)} / ?dZ {z})) +
- (\<Sum> (x, y, z) \<in> (X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M).
- ?dXYZ {(x, y, z)} * log b (?dXYZ {(x, y, z)} / ?dYZ {(y, z)}))"
- using finite_space
- apply (auto simp: setsum_cartesian_product[symmetric] setsum_negf[symmetric]
- setsum_addf[symmetric] diff_minus
- intro!: setsum_cong[OF refl])
- apply (subst split_measure)
- apply (simp add: If_mult_distr setsum_cases sets distribution_log_split[OF assms, of X])
- apply (subst add_commute)
- by (simp add: setsum_subtractf setsum_negf field_simps setsum_right_distrib[symmetric] sets_eq_Pow)
+ assume *: "0 < ?dXYZ {(fst x, fst (snd x), snd (snd x))}"
+ thus "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+ by (auto intro!: divide_pos_pos mult_pos_pos
+ intro: distribution_order(6) distribution_mono_gt_0)
+ qed (simp_all add: setsum_cartesian_product' sum_over_space_distrib setsum_distribution finite_space)
+ also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>Z`space M. ?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_distribution
+ intro!: setsum_cong)
+ finally show ?thesis
+ unfolding conditional_mutual_information_eq sum_over_space_distrib by simp
qed
+
definition (in prob_space)
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
-abbreviation (in finite_prob_space)
- finite_conditional_entropy ("\<H>\<^bsub>_\<^esub>'(_ | _')") where
- "\<H>\<^bsub>b\<^esub>(X | Y) \<equiv> conditional_entropy b
+abbreviation (in finite_information_space)
+ finite_conditional_entropy ("\<H>'(_ | _')") where
+ "\<H>(X | Y) \<equiv> conditional_entropy b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
-abbreviation (in finite_prob_space)
- finite_conditional_entropy_2 ("\<H>'(_ | _')") where
- "\<H>(X | Y) \<equiv> \<H>\<^bsub>2\<^esub>(X | Y)"
+lemma (in finite_information_space) conditional_entropy_positive:
+ "0 \<le> \<H>(X | Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive .
-lemma (in finite_prob_space) finite_conditional_entropy_reduce:
- assumes "1 < b"
- shows "\<H>\<^bsub>b\<^esub>(X | Z) =
+lemma (in finite_information_space) conditional_entropy_eq:
+ "\<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}))"
proof -
have *: "\<And>x y z. (\<lambda>x. (X x, X x, Z x)) -` {(x, y, z)} = (if x = y then (\<lambda>x. (X x, Z x)) -` {(x, z)} else {})" by auto
show ?thesis
- unfolding finite_conditional_mutual_information_reduce[OF assms]
- conditional_entropy_def joint_distribution_def distribution_def *
+ unfolding conditional_mutual_information_eq_sum
+ conditional_entropy_def distribution_def *
by (auto intro!: setsum_0')
qed
-lemma (in finite_prob_space) finite_mutual_information_eq_entropy_conditional_entropy:
- assumes "1 < b" shows "\<I>\<^bsub>b\<^esub>(X ; Z) = \<H>\<^bsub>b\<^esub>(X) - \<H>\<^bsub>b\<^esub>(X | Z)" (is "mutual_information b ?X ?Z X Z = _")
- unfolding finite_mutual_information_reduce
- finite_entropy_reduce[OF assms]
- finite_conditional_entropy_reduce[OF assms]
- joint_distribution diff_minus_eq_add
+lemma (in finite_information_space) mutual_information_eq_entropy_conditional_entropy:
+ "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
+ unfolding mutual_information_eq entropy_eq conditional_entropy_eq
using finite_space
- apply (auto simp add: setsum_addf[symmetric] setsum_subtractf
- setsum_Sigma[symmetric] distribution_log_split[OF assms] setsum_negf[symmetric]
- intro!: setsum_cong[OF refl])
- apply (simp add: setsum_negf setsum_left_distrib[symmetric])
-proof (rule disjI2)
- let ?dX = "distribution X"
- and ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
+ by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product'
+ setsum_left_distrib[symmetric] setsum_addf setsum_distribution)
- fix x assume "x \<in> space M"
- have "\<And>z. (\<lambda>x. (X x, Z x)) -` {(X x, z)} \<inter> space M = (X -` {X x} \<inter> space M) \<inter> (Z -` {z} \<inter> space M)" by auto
- thus "(\<Sum>z\<in>Z ` space M. distribution (\<lambda>x. (X x, Z x)) {(X x, z)}) = distribution X {X x}"
- unfolding distribution_def
- apply (subst prob_real_sum_image_fn[where e="X -` {X x} \<inter> space M" and s = "Z`space M" and f="\<lambda>z. Z -` {z} \<inter> space M"])
- using finite_space sets_eq_Pow by auto
+lemma (in finite_information_space) conditional_entropy_less_eq_entropy:
+ "\<H>(X | Z) \<le> \<H>(X)"
+proof -
+ have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy .
+ with mutual_information_positive[of X Z] entropy_positive[of X]
+ show ?thesis by auto
qed
(* -------------Entropy of a RV with a certain event is zero---------------- *)
-lemma (in finite_prob_space) finite_entropy_certainty_eq_0:
- assumes "x \<in> X ` space M" and "distribution X {x} = 1" and "b > 1"
- shows "\<H>\<^bsub>b\<^esub>(X) = 0"
+lemma (in finite_information_space) finite_entropy_certainty_eq_0:
+ assumes "x \<in> X ` space M" and "distribution X {x} = 1"
+ shows "\<H>(X) = 0"
proof -
interpret X: finite_prob_space "\<lparr> space = X ` space M,
sets = Pow (X ` space M),
- measure = distribution X\<rparr>" by (rule finite_prob_space)
+ measure = distribution X\<rparr>" by (rule finite_prob_space_of_images)
have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
using X.measure_compl[of "{x}"] assms by auto
@@ -694,366 +606,18 @@
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 finite_entropy_reduce[OF `b > 1`] by (auto simp: y fi)
+ show ?thesis unfolding entropy_eq by (auto simp: y fi)
qed
(* --------------- upper bound on entropy for a rv ------------------------- *)
-definition convex_set :: "real set \<Rightarrow> bool"
-where
- "convex_set C \<equiv> (\<forall> x y \<mu>. x \<in> C \<and> y \<in> C \<and> 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow> \<mu> * x + (1 - \<mu>) * y \<in> C)"
-
-lemma pos_is_convex:
- shows "convex_set {0 <..}"
-unfolding convex_set_def
-proof safe
- fix x y \<mu> :: real
- assume asms: "\<mu> \<ge> 0" "\<mu> \<le> 1" "x > 0" "y > 0"
- { assume "\<mu> = 0"
- hence "\<mu> * x + (1 - \<mu>) * y = y" by simp
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms by simp }
- moreover
- { assume "\<mu> = 1"
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms by simp }
- moreover
- { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
- hence "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms
- apply (subst add_nonneg_pos[of "\<mu> * x" "(1 - \<mu>) * y"])
- using real_mult_order by auto fastsimp }
- ultimately show "\<mu> * x + (1 - \<mu>) * y > 0" using assms by blast
-qed
-
-definition convex_fun :: "(real \<Rightarrow> real) \<Rightarrow> real set \<Rightarrow> bool"
-where
- "convex_fun f C \<equiv> (\<forall> x y \<mu>. convex_set C \<and> (x \<in> C \<and> y \<in> C \<and> 0 \<le> \<mu> \<and> \<mu> \<le> 1
- \<longrightarrow> f (\<mu> * x + (1 - \<mu>) * y) \<le> \<mu> * f x + (1 - \<mu>) * f y))"
-
-lemma pos_convex_function:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_set C"
- assumes leq: "\<And> x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
- shows "convex_fun f C"
-unfolding convex_fun_def
-using assms
-proof safe
- fix x y \<mu> :: real
- let ?x = "\<mu> * x + (1 - \<mu>) * y"
- assume asm: "convex_set C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- hence "1 - \<mu> \<ge> 0" by auto
- hence xpos: "?x \<in> C" using asm unfolding convex_set_def by auto
- have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x)
- \<ge> \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
- using add_mono[OF mult_mono1[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`]
- mult_mono1[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
- hence "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
- by (auto simp add:field_simps)
- thus "\<mu> * f x + (1 - \<mu>) * f y \<ge> f ?x" by simp
-qed
-
-lemma atMostAtLeast_subset_convex:
- assumes "convex_set C"
- assumes "x \<in> C" "y \<in> C" "x < y"
- shows "{x .. y} \<subseteq> C"
-proof safe
- fix z assume zasm: "z \<in> {x .. y}"
- { assume asm: "x < z" "z < y"
- let "?\<mu>" = "(y - z) / (y - x)"
- have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add:field_simps)
- hence comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
- using assms[unfolded convex_set_def] by blast
- have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
- by (auto simp add:field_simps)
- also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
- using assms unfolding add_divide_distrib by (auto simp:field_simps)
- also have "\<dots> = z"
- using assms by (auto simp:field_simps)
- finally have "z \<in> C"
- using comb by auto } note less = this
- show "z \<in> C" using zasm less assms
- unfolding atLeastAtMost_iff le_less by auto
-qed
-
-lemma f''_imp_f':
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_set C"
- assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- assumes "x \<in> C" "y \<in> C"
- shows "f' x * (y - x) \<le> f y - f x"
-using assms
-proof -
- { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "y > x"
- hence ge: "y - x > 0" "y - x \<ge> 0" by auto
- from asm have le: "x - y < 0" "x - y \<le> 0" by auto
- then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
- using subsetD[OF atMostAtLeast_subset_convex[OF `convex_set C` `x \<in> C` `y \<in> C` `x < y`],
- THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
- by auto
- hence "z1 \<in> C" using atMostAtLeast_subset_convex
- `convex_set C` `x \<in> C` `y \<in> C` `x < y` by fastsimp
- from z1 have z1': "f x - f y = (x - y) * f' z1"
- by (simp add:field_simps)
- obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
- using subsetD[OF atMostAtLeast_subset_convex[OF `convex_set C` `x \<in> C` `z1 \<in> C` `x < z1`],
- THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
- using subsetD[OF atMostAtLeast_subset_convex[OF `convex_set C` `z1 \<in> C` `y \<in> C` `z1 < y`],
- THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
- using asm z1' by auto
- also have "\<dots> = (y - z1) * f'' z3" using z3 by auto
- finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" by simp
- have A': "y - z1 \<ge> 0" using z1 by auto
- have "z3 \<in> C" using z3 asm atMostAtLeast_subset_convex
- `convex_set C` `x \<in> C` `z1 \<in> C` `x < z1` by fastsimp
- hence B': "f'' z3 \<ge> 0" using assms by auto
- from A' B' have "(y - z1) * f'' z3 \<ge> 0" using mult_nonneg_nonneg by auto
- from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0" by auto
- from mult_right_mono_neg[OF this le(2)]
- have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
- unfolding diff_def using real_add_mult_distrib by auto
- hence "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
- hence res: "f' y * (x - y) \<le> f x - f y" by auto
- have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
- using asm z1 by auto
- also have "\<dots> = (z1 - x) * f'' z2" using z2 by auto
- finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" by simp
- have A: "z1 - x \<ge> 0" using z1 by auto
- have "z2 \<in> C" using z2 z1 asm atMostAtLeast_subset_convex
- `convex_set C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastsimp
- hence B: "f'' z2 \<ge> 0" using assms by auto
- from A B have "(z1 - x) * f'' z2 \<ge> 0" using mult_nonneg_nonneg by auto
- from cool this have "(f y - f x) / (y - x) - f' x \<ge> 0" by auto
- from mult_right_mono[OF this ge(2)]
- have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
- unfolding diff_def using real_add_mult_distrib by auto
- hence "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
- hence "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
- using res by auto } note less_imp = this
- { fix x y :: real assume "x \<in> C" "y \<in> C" "x \<noteq> y"
- hence"f y - f x \<ge> f' x * (y - x)"
- unfolding neq_iff apply safe
- using less_imp by auto } note neq_imp = this
- moreover
- { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "x = y"
- hence "f y - f x \<ge> f' x * (y - x)" by auto }
- ultimately show ?thesis using assms by blast
-qed
-
-lemma f''_ge0_imp_convex:
- fixes f :: "real \<Rightarrow> real"
- assumes conv: "convex_set C"
- assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- shows "convex_fun f C"
-using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function by fastsimp
-
-lemma minus_log_convex:
- fixes b :: real
- assumes "b > 1"
- shows "convex_fun (\<lambda> x. - log b x) {0 <..}"
-proof -
- have "\<And> z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" using DERIV_log by auto
- hence f': "\<And> z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
- using DERIV_minus by auto
- have "\<And> z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
- using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
- from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
- have "\<And> z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
- by auto
- hence f''0: "\<And> z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
- unfolding inverse_eq_divide by (auto simp add:real_mult_assoc)
- have f''_ge0: "\<And> z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
- using `b > 1` by (auto intro!:less_imp_le simp add:divide_pos_pos[of 1] real_mult_order)
- from f''_ge0_imp_convex[OF pos_is_convex,
- unfolded greaterThan_iff, OF f' f''0 f''_ge0]
- show ?thesis by auto
-qed
-
-lemma setsum_nonneg_0:
- fixes f :: "'a \<Rightarrow> real"
- assumes "finite s"
- assumes "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
- assumes "(\<Sum> i \<in> s. f i) = 0"
- assumes "i \<in> s"
- shows "f i = 0"
-proof -
- { assume asm: "f i > 0"
- from assms have "\<forall> j \<in> s - {i}. f j \<ge> 0" by auto
- from setsum_nonneg[of "s - {i}" f, OF this]
- have "(\<Sum> j \<in> s - {i}. f j) \<ge> 0" by simp
- hence "(\<Sum> j \<in> s - {i}. f j) + f i > 0" using asm by auto
- from this setsum.remove[of s i f, OF `finite s` `i \<in> s`]
- have "(\<Sum> j \<in> s. f j) > 0" by auto
- hence "False" using assms by auto }
- thus ?thesis using assms by fastsimp
-qed
-
-lemma setsum_nonneg_leq_1:
- fixes f :: "'a \<Rightarrow> real"
- assumes "finite s"
- assumes "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
- assumes "(\<Sum> i \<in> s. f i) = 1"
- assumes "i \<in> s"
- shows "f i \<le> 1"
-proof -
- { assume asm: "f i > 1"
- from assms have "\<forall> j \<in> s - {i}. f j \<ge> 0" by auto
- from setsum_nonneg[of "s - {i}" f, OF this]
- have "(\<Sum> j \<in> s - {i}. f j) \<ge> 0" by simp
- hence "(\<Sum> j \<in> s - {i}. f j) + f i > 1" using asm by auto
- from this setsum.remove[of s i f, OF `finite s` `i \<in> s`]
- have "(\<Sum> j \<in> s. f j) > 1" by auto
- hence "False" using assms by auto }
- thus ?thesis using assms by fastsimp
-qed
-
-lemma convex_set_setsum:
- assumes "finite s" "s \<noteq> {}"
- assumes "convex_set C"
- 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> C"
- shows "(\<Sum> j \<in> s. a j * y j) \<in> C"
-using assms
-proof (induct s arbitrary:a rule:finite_ne_induct)
- case (singleton i) note asms = this
- hence "a i = 1" by auto
- thus ?case using asms by auto
-next
- case (insert i s) note asms = this
- { assume "a i = 1"
- hence "(\<Sum> j \<in> s. a j) = 0"
- using asms by auto
- hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
- using setsum_nonneg_0 asms by fastsimp
- hence ?case using asms by auto }
- moreover
- { assume asm: "a i \<noteq> 1"
- from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
- have fis: "finite (insert i s)" using asms by auto
- hence ai1: "a i \<le> 1" using setsum_nonneg_leq_1[of "insert i s" a] asms by simp
- hence "a i < 1" using asm by auto
- hence i0: "1 - a i > 0" by auto
- let "?a j" = "a j / (1 - a i)"
- { fix j assume "j \<in> s"
- hence "?a j \<ge> 0"
- using i0 asms divide_nonneg_pos
- by fastsimp } note a_nonneg = this
- have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
- hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
- hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
- hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
- from this asms
- have "(\<Sum>j\<in>s. ?a j * y j) \<in> C" using a_nonneg by fastsimp
- hence "a i * y i + (1 - a i) * (\<Sum> j \<in> s. ?a j * y j) \<in> C"
- using asms[unfolded convex_set_def, rule_format] yai ai1 by auto
- hence "a i * y i + (\<Sum> j \<in> s. (1 - a i) * (?a j * y j)) \<in> C"
- using mult_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j * y j" s] by auto
- hence "a i * y i + (\<Sum> j \<in> s. a j * y j) \<in> C" using i0 by auto
- hence ?case using setsum.insert asms by auto }
- ultimately show ?case by auto
-qed
-
-lemma convex_fun_setsum:
- fixes a :: "'a \<Rightarrow> real"
- assumes "finite s" "s \<noteq> {}"
- assumes "convex_fun f C"
- 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> C"
- shows "f (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
-using assms
-proof (induct s arbitrary:a rule:finite_ne_induct)
- case (singleton i)
- hence ai: "a i = 1" by auto
- thus ?case by auto
-next
- case (insert i s) note asms = this
- hence "convex_fun f C" by simp
- from this[unfolded convex_fun_def, rule_format]
- have conv: "\<And> x y \<mu>. \<lbrakk>x \<in> C; y \<in> C; 0 \<le> \<mu>; \<mu> \<le> 1\<rbrakk>
- \<Longrightarrow> f (\<mu> * x + (1 - \<mu>) * y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by simp
- { assume "a i = 1"
- hence "(\<Sum> j \<in> s. a j) = 0"
- using asms by auto
- hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
- using setsum_nonneg_0 asms by fastsimp
- hence ?case using asms by auto }
- moreover
- { assume asm: "a i \<noteq> 1"
- from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
- have fis: "finite (insert i s)" using asms by auto
- hence ai1: "a i \<le> 1" using setsum_nonneg_leq_1[of "insert i s" a] asms by simp
- hence "a i < 1" using asm by auto
- hence i0: "1 - a i > 0" by auto
- let "?a j" = "a j / (1 - a i)"
- { fix j assume "j \<in> s"
- hence "?a j \<ge> 0"
- using i0 asms divide_nonneg_pos
- by fastsimp } note a_nonneg = this
- have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
- hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
- hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
- hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
- have "convex_set C" using asms unfolding convex_fun_def by auto
- hence asum: "(\<Sum> j \<in> s. ?a j * y j) \<in> C"
- using asms convex_set_setsum[OF `finite s` `s \<noteq> {}`
- `convex_set C` a1 a_nonneg] by auto
- have asum_le: "f (\<Sum> j \<in> s. ?a j * y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
- using a_nonneg a1 asms by blast
- have "f (\<Sum> j \<in> insert i s. a j * y j) = f ((\<Sum> j \<in> s. a j * y j) + a i * y i)"
- using setsum.insert[of s i "\<lambda> j. a j * y j", OF `finite s` `i \<notin> s`] asms
- by (auto simp only:add_commute)
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. a j * y j) / (1 - a i) + a i * y i)"
- using i0 by auto
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. a j * y j / (1 - a i)) + a i * y i)"
- unfolding divide.setsum[of "\<lambda> j. a j * y j" s "1 - a i", symmetric] by auto
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. ?a j * y j) + a i * y i)" by auto
- also have "\<dots> \<le> (1 - a i) * f ((\<Sum> j \<in> s. ?a j * y j)) + a i * f (y i)"
- using conv[of "y i" "(\<Sum> j \<in> s. ?a j * y j)" "a i", OF yai(1) asum yai(2) ai1]
- by (auto simp only:add_commute)
- also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
- using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
- OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
- also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
- unfolding mult_right.setsum[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)" using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))" using asms by auto
- finally have "f (\<Sum> j \<in> insert i s. a j * y j) \<le> (\<Sum> j \<in> insert i s. a j * f (y j))"
- by simp }
- ultimately show ?case by auto
-qed
-
-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_fun (\<lambda> x. - log b x) {0 <..}"
- 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_fun_setsum assms by blast
- thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
-qed
-
-lemma (in finite_prob_space) finite_entropy_le_card:
- assumes "1 < b"
- shows "\<H>\<^bsub>b\<^esub>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
+lemma (in finite_information_space) finite_entropy_le_card:
+ "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
proof -
interpret X: finite_prob_space "\<lparr>space = X ` space M,
sets = Pow (X ` space M),
measure = distribution X\<rparr>"
- using finite_prob_space by auto
+ using finite_prob_space_of_images by auto
+
have triv: "\<And> x. (if distribution X {x} \<noteq> 0 then distribution X {x} else 0) = distribution X {x}"
by auto
hence sum1: "(\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x}) = 1"
@@ -1085,7 +649,7 @@
also have "\<dots> = (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (inverse (distribution X {x}))
else 0)"
- using log_inverse `1 < b` X.positive[of "{x}"] asm by auto
+ using log_inverse b_gt_1 X.positive[of "{x}"] asm by auto
finally have "- distribution X {x} * log b (distribution X {x})
= (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (inverse (distribution X {x}))
@@ -1101,7 +665,7 @@
unfolding setsum_restrict_set[OF finite_imageI[OF finite_space, of X]] by auto
also have "\<dots> \<le> log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
distribution X {x} * (inverse (distribution X {x})))"
- apply (subst log_setsum[OF _ _ `b > 1` sum1,
+ apply (subst log_setsum[OF _ _ b_gt_1 sum1,
unfolded greaterThan_iff, OF _ _ _]) using pos sets_eq_Pow
X.finite_space assms X.positive not_empty by auto
also have "\<dots> = log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 1)"
@@ -1110,7 +674,7 @@
by auto
finally have "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
\<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" by simp
- thus ?thesis unfolding finite_entropy_reduce[OF assms] real_eq_of_nat by auto
+ thus ?thesis unfolding entropy_eq real_eq_of_nat by auto
qed
(* --------------- entropy is maximal for a uniform rv --------------------- *)
@@ -1140,34 +704,31 @@
by (auto simp:field_simps)
qed
-lemma (in finite_prob_space) finite_entropy_uniform_max:
- assumes "b > 1"
+lemma (in finite_information_space) finite_entropy_uniform_max:
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>\<^bsub>b\<^esub>(X) = log b (real (card (X ` space M)))"
+ shows "\<H>(X) = log b (real (card (X ` space M)))"
proof -
interpret X: finite_prob_space "\<lparr>space = X ` space M,
sets = Pow (X ` space M),
measure = distribution X\<rparr>"
- using finite_prob_space by auto
+ using finite_prob_space_of_images by auto
+
{ fix x assume xasm: "x \<in> X ` space M"
hence card_gt0: "real (card (X ` space M)) > 0"
using card_gt_0_iff X.finite_space by auto
from xasm have "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = distribution X {x}"
using assms by blast
hence "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
- = - (\<Sum> y \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
- by auto
- also have "\<dots> = - real_of_nat (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
- by auto
+ = - real (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
+ unfolding real_eq_of_nat by auto
also have "\<dots> = - real (card (X ` space M)) * (1 / real (card (X ` space M))) * log b (1 / real (card (X ` space M)))"
- unfolding real_eq_of_nat[symmetric]
- by (auto simp: X.uniform_prob[simplified, OF xasm assms(2)])
+ by (auto simp: X.uniform_prob[simplified, OF xasm assms])
also have "\<dots> = log b (real (card (X ` space M)))"
unfolding inverse_eq_divide[symmetric]
- using card_gt0 log_inverse `b > 1`
+ using card_gt0 log_inverse b_gt_1
by (auto simp add:field_simps card_gt0)
finally have ?thesis
- unfolding finite_entropy_reduce[OF `b > 1`] by auto }
+ unfolding entropy_eq by auto }
moreover
{ assume "X ` space M = {}"
hence "distribution X (X ` space M) = 0"
@@ -1176,4 +737,199 @@
ultimately show ?thesis by auto
qed
+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 finite_information_space) entropy_partition:
+ assumes svi: "subvimage (space M) X P"
+ shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
+proof -
+ have "(\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) =
+ (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
+ joint_distribution X P {(x, y)} * log b (joint_distribution X P {(x, y)}))"
+ proof (subst setsum_image_split[OF svi],
+ safe intro!: finite_imageI finite_space setsum_mono_zero_cong_left imageI)
+ fix p x assume in_space: "p \<in> space M" "x \<in> space M"
+ assume "joint_distribution X P {(X x, P p)} * log b (joint_distribution X P {(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 "distribution X {X x} * log b (distribution X {X x}) =
+ joint_distribution X P {(X x, P p)} *
+ log b (joint_distribution X P {(X x, P p)})"
+ by (auto simp: distribution_def)
+ qed
+ thus ?thesis
+ unfolding entropy_eq conditional_entropy_eq
+ by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
+ setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
+qed
+
+corollary (in finite_information_space) entropy_data_processing:
+ "\<H>(f \<circ> X) \<le> \<H>(X)"
+ by (subst (2) entropy_partition[of _ "f \<circ> X"]) (auto intro: conditional_entropy_positive)
+
+lemma (in prob_space) distribution_cong:
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
+ shows "distribution X = distribution Y"
+ unfolding distribution_def expand_fun_eq
+ using assms by (auto intro!: arg_cong[where f=prob])
+
+lemma (in prob_space) joint_distribution_cong:
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
+ shows "joint_distribution X Y = joint_distribution X' Y'"
+ unfolding distribution_def expand_fun_eq
+ using assms by (auto intro!: arg_cong[where f=prob])
+
+lemma image_cong:
+ "\<lbrakk> \<And>x. x \<in> S \<Longrightarrow> X x = X' x \<rbrakk> \<Longrightarrow> X ` S = X' ` S"
+ by (auto intro!: image_eqI)
+
+lemma (in finite_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')"
+proof -
+ have "X ` space M = X' ` space M" using X by (rule image_cong)
+ moreover have "Y ` space M = Y' ` space M" using Y by (rule image_cong)
+ ultimately show ?thesis
+ unfolding mutual_information_eq
+ using
+ assms[THEN distribution_cong]
+ joint_distribution_cong[OF assms]
+ by (auto intro!: setsum_cong)
+qed
+
+corollary (in finite_information_space) entropy_of_inj:
+ assumes "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 .
+next
+ 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 assms])
+ also have "... \<le> \<H>(f \<circ> X)"
+ using entropy_data_processing .
+ finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
+qed
+
end