--- a/src/HOL/Probability/Information.thy Wed Dec 01 06:50:54 2010 -0800
+++ b/src/HOL/Probability/Information.thy Wed Dec 01 19:20:30 2010 +0100
@@ -1,5 +1,5 @@
theory Information
-imports Probability_Space Product_Measure Convex Radon_Nikodym
+imports Probability_Space Convex Lebesgue_Measure
begin
lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
@@ -12,43 +12,6 @@
"(\<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
-lemma real_of_pinfreal_inverse[simp]:
- fixes X :: pinfreal
- shows "real (inverse X) = 1 / real X"
- by (cases X) (auto simp: inverse_eq_divide)
-
-lemma (in finite_prob_space) finite_product_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M)\<rparr>
- (joint_distribution X Y)"
- (is "finite_prob_space ?S _")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images)
- have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
- thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
- by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
-qed
-
-lemma (in finite_prob_space) finite_measure_space_prod:
- assumes X: "finite_measure_space MX (distribution X)"
- assumes Y: "finite_measure_space MY (distribution Y)"
- shows "finite_measure_space (prod_measure_space MX MY) (joint_distribution X Y)"
- (is "finite_measure_space ?M ?D")
-proof (intro finite_measure_spaceI)
- interpret X: finite_measure_space MX "distribution X" by fact
- interpret Y: finite_measure_space MY "distribution Y" by fact
- note finite_measure_space.finite_prod_measure_space[OF X Y, simp]
- show "finite (space ?M)" using X.finite_space Y.finite_space by auto
- show "joint_distribution X Y {} = 0" by simp
- show "sets ?M = Pow (space ?M)" by simp
- { fix x show "?D (space ?M) \<noteq> \<omega>" by (rule distribution_finite) }
- { fix A B assume "A \<subseteq> space ?M" "B \<subseteq> space ?M" "A \<inter> B = {}"
- have *: "(\<lambda>t. (X t, Y t)) -` (A \<union> B) \<inter> space M =
- (\<lambda>t. (X t, Y t)) -` A \<inter> space M \<union> (\<lambda>t. (X t, Y t)) -` B \<inter> space M"
- by auto
- show "?D (A \<union> B) = ?D A + ?D B" unfolding distribution_def *
- apply (rule measure_additive[symmetric])
- using `A \<inter> B = {}` by (auto simp: sets_eq_Pow) }
-qed
-
section "Convex theory"
lemma log_setsum:
@@ -148,82 +111,48 @@
qed
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
+ "((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 finite_information_space = finite_prob_space +
+locale information_space = prob_space +
fixes b :: real assumes b_gt_1: "1 < b"
-context finite_information_space
+context information_space
begin
-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")
+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 -
- 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
+ { 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
-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 real_pinfreal_nonneg},
- @{thm real_distribution_divide_pos_pos},
- @{thm real_distribution_mult_inverse_pos_pos},
- @{thm real_distribution_mult_pos_pos}]
-
- 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
+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)
- 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
+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
- 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
-*}
+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)
-simproc_setup mult_log ("real (distribution X x) * log b (A * B)" |
- "real (distribution X x) * log b (A / B)") = {* K mult_log_simproc *}
+lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
end
@@ -236,15 +165,49 @@
"KL_divergence b M \<mu> \<nu> =
measure_space.integral M \<mu> (\<lambda>x. log b (real (sigma_finite_measure.RN_deriv M \<nu> \<mu> x)))"
+lemma (in sigma_finite_measure) KL_divergence_cong:
+ assumes "measure_space M \<nu>"
+ and cong: "\<And>A. A \<in> sets M \<Longrightarrow> \<mu>' A = \<mu> A" "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
+ shows "KL_divergence b M \<nu>' \<mu>' = KL_divergence b M \<nu> \<mu>"
+proof -
+ interpret \<nu>: measure_space M \<nu> by fact
+ show ?thesis
+ unfolding KL_divergence_def
+ using RN_deriv_cong[OF cong, of "\<lambda>A. A"]
+ by (simp add: cong \<nu>.integral_cong_measure[OF cong(2)])
+qed
+
+lemma (in sigma_finite_measure) KL_divergence_vimage:
+ assumes f: "bij_betw f S (space M)"
+ assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
+ shows "KL_divergence b (vimage_algebra S f) (\<lambda>A. \<nu> (f ` A)) (\<lambda>A. \<mu> (f ` A)) = KL_divergence b M \<nu> \<mu>"
+ (is "KL_divergence b ?M ?\<nu> ?\<mu> = _")
+proof -
+ interpret \<nu>: measure_space M \<nu> by fact
+ interpret v: measure_space ?M ?\<nu>
+ using f by (rule \<nu>.measure_space_isomorphic)
+
+ let ?RN = "sigma_finite_measure.RN_deriv ?M ?\<mu> ?\<nu>"
+ from RN_deriv_vimage[OF f \<nu>]
+ have *: "\<nu>.almost_everywhere (\<lambda>x. ?RN (the_inv_into S f x) = RN_deriv \<nu> x)"
+ by (rule absolutely_continuous_AE[OF \<nu>])
+
+ show ?thesis
+ unfolding KL_divergence_def \<nu>.integral_vimage_inv[OF f]
+ apply (rule \<nu>.integral_cong_AE)
+ apply (rule \<nu>.AE_mp[OF *])
+ apply (rule \<nu>.AE_cong)
+ apply simp
+ done
+qed
+
lemma (in finite_measure_space) KL_divergence_eq_finite:
assumes v: "finite_measure_space M \<nu>"
- assumes ac: "\<forall>x\<in>space M. \<mu> {x} = 0 \<longrightarrow> \<nu> {x} = 0"
+ assumes ac: "absolutely_continuous \<nu>"
shows "KL_divergence b M \<nu> \<mu> = (\<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 \<nu> by fact
have ms: "measure_space M \<nu>" by fact
- have ac: "absolutely_continuous \<nu>"
- using ac by (auto intro!: absolutely_continuousI[OF v])
show "(\<Sum>x \<in> space M. log b (real (RN_deriv \<nu> x)) * real (\<nu> {x})) = ?sum"
using RN_deriv_finite_measure[OF ms ac]
by (auto intro!: setsum_cong simp: field_simps real_of_pinfreal_mult[symmetric])
@@ -252,32 +215,27 @@
lemma (in finite_prob_space) KL_divergence_positive_finite:
assumes v: "finite_prob_space M \<nu>"
- assumes ac: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
+ assumes ac: "absolutely_continuous \<nu>"
and "1 < b"
shows "0 \<le> KL_divergence b M \<nu> \<mu>"
proof -
interpret v: finite_prob_space M \<nu> using v .
-
- have *: "space M \<noteq> {}" using not_empty by simp
+ have ms: "finite_measure_space M \<nu>" by default
- hence "- (KL_divergence b M \<nu> \<mu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
- proof (subst KL_divergence_eq_finite)
- show "finite_measure_space M \<nu>" by fact
+ have "- (KL_divergence b M \<nu> \<mu>) \<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
- show "\<forall>x\<in>space M. \<mu> {x} = 0 \<longrightarrow> \<nu> {x} = 0" using ac by auto
- show "- (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x}))) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
- proof (safe intro!: log_setsum_divide *)
- 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
-
- fix x assume x: "x \<in> space M"
- { assume "0 < real (\<nu> {x})"
- hence "\<mu> {x} \<noteq> 0" using ac[OF x] by auto
- thus "0 < prob {x}" using finite_measure[of "{x}"] sets_eq_Pow x
- by (cases "\<mu> {x}") simp_all }
- qed auto
- qed
+ 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 < prob {x}" using finite_measure[of "{x}"] x by auto }
+ qed auto
thus "0 \<le> KL_divergence b M \<nu> \<mu>" using finite_sum_over_space_eq_1 by simp
qed
@@ -285,174 +243,175 @@
definition (in prob_space)
"mutual_information b S T X Y =
- KL_divergence b (prod_measure_space S T)
+ KL_divergence b (sigma (pair_algebra S T))
(joint_distribution X Y)
- (prod_measure S (distribution X) T (distribution Y))"
+ (pair_sigma_finite.pair_measure S (distribution X) T (distribution Y))"
-abbreviation (in finite_information_space)
- finite_mutual_information ("\<I>'(_ ; _')") where
+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) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
-lemma (in finite_information_space) mutual_information_generic_eq:
- assumes MX: "finite_measure_space MX (distribution X)"
- assumes MY: "finite_measure_space MY (distribution Y)"
- shows "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
- real (joint_distribution X Y {(x,y)}) *
- log b (real (joint_distribution X Y {(x,y)}) /
- (real (distribution X {x}) * real (distribution Y {y}))))"
+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 (sigma (pair_algebra S T))
+ (pair_sigma_finite.pair_measure S (distribution X) T (distribution Y)) (joint_distribution X Y)"
+ shows "mutual_information b S T X Y = mutual_information b T S Y X"
proof -
- let ?P = "prod_measure_space MX MY"
- let ?\<mu> = "prod_measure MX (distribution X) MY (distribution Y)"
- let ?\<nu> = "joint_distribution X Y"
- interpret X: finite_measure_space MX "distribution X" by fact
- moreover interpret Y: finite_measure_space MY "distribution Y" by fact
- have fms: "finite_measure_space MX (distribution X)"
- "finite_measure_space MY (distribution Y)" by fact+
- have fms_P: "finite_measure_space ?P ?\<mu>"
- by (rule X.finite_measure_space_finite_prod_measure) fact
- then interpret P: finite_measure_space ?P ?\<mu> .
- have fms_P': "finite_measure_space ?P ?\<nu>"
- 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 sigma_def)
- then interpret P': finite_measure_space ?P ?\<nu> .
- { fix x assume "x \<in> space ?P"
- hence in_MX: "{fst x} \<in> sets MX" "{snd x} \<in> sets MY" using X.sets_eq_Pow Y.sets_eq_Pow
- by (auto simp: prod_measure_space_def)
- assume "?\<mu> {x} = 0"
- with X.finite_prod_measure_times[OF fms(2), of "{fst x}" "{snd 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
+ interpret P: prob_space "sigma (pair_algebra S T)" "joint_distribution X Y"
+ using random_variable_pairI[OF X Y] by (rule distribution_prob_space)
+ interpret Q: prob_space "sigma (pair_algebra T S)" "joint_distribution Y X"
+ using random_variable_pairI[OF Y X] by (rule distribution_prob_space)
+ interpret X: prob_space S "distribution X" using X by (rule distribution_prob_space)
+ interpret Y: prob_space T "distribution Y" using Y by (rule distribution_prob_space)
+ interpret ST: pair_sigma_finite S "distribution X" T "distribution Y" by default
+ interpret TS: pair_sigma_finite T "distribution Y" S "distribution X" by default
+
+ have ST: "measure_space (sigma (pair_algebra S T)) (joint_distribution X Y)" by default
+ have TS: "measure_space (sigma (pair_algebra T S)) (joint_distribution Y X)" by default
+
+ have bij_ST: "bij_betw (\<lambda>(x, y). (y, x)) (space (sigma (pair_algebra S T))) (space (sigma (pair_algebra T S)))"
+ by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
+ have bij_TS: "bij_betw (\<lambda>(x, y). (y, x)) (space (sigma (pair_algebra T S))) (space (sigma (pair_algebra S T)))"
+ by (auto intro!: inj_onI simp: space_pair_algebra bij_betw_def)
+
+ { fix A
+ have "joint_distribution X Y ((\<lambda>(x, y). (y, x)) ` A) = joint_distribution Y X A"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>]) }
+ note jd_commute = this
+
+ { fix A assume A: "A \<in> sets (sigma (pair_algebra T S))"
+ have *: "\<And>x y. indicator ((\<lambda>(x, y). (y, x)) ` A) (x, y) = (indicator A (y, x) :: pinfreal)"
+ unfolding indicator_def by auto
+ have "ST.pair_measure ((\<lambda>(x, y). (y, x)) ` A) = TS.pair_measure A"
+ unfolding ST.pair_measure_def TS.pair_measure_def
+ using A by (auto simp add: TS.Fubini[symmetric] *) }
+ note pair_measure_commute = this
+
show ?thesis
- unfolding Let_def mutual_information_def
- using measure_0 fms_P fms_P' MX MY P.absolutely_continuous_def
- by (subst P.KL_divergence_eq_finite)
- (auto simp add: prod_measure_space_def prod_measure_times_finite
- finite_prob_space_eq setsum_cartesian_product' real_of_pinfreal_mult[symmetric])
+ unfolding mutual_information_def
+ unfolding ST.KL_divergence_vimage[OF bij_TS ST ac, symmetric]
+ unfolding space_sigma space_pair_algebra jd_commute
+ unfolding ST.pair_sigma_algebra_swap[symmetric]
+ by (simp cong: TS.KL_divergence_cong[OF TS] add: pair_measure_commute)
qed
-lemma (in finite_information_space)
- assumes MX: "finite_prob_space MX (distribution X)"
- assumes MY: "finite_prob_space MY (distribution Y)"
- and X_space: "X ` space M \<subseteq> space MX" and Y_space: "Y ` space M \<subseteq> space MY"
- shows mutual_information_eq_generic:
+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 (sigma (pair_algebra S T))
+ (pair_sigma_finite.pair_measure S (distribution X) T (distribution Y)) (joint_distribution X Y)"
+proof -
+ interpret X: finite_prob_space S "distribution X" using X by (rule distribution_finite_prob_space)
+ interpret Y: finite_prob_space T "distribution Y" using Y by (rule distribution_finite_prob_space)
+ interpret XY: pair_finite_prob_space S "distribution X" T "distribution Y" by default
+ interpret P: finite_prob_space XY.P "joint_distribution X Y"
+ using assms by (intro joint_distribution_finite_prob_space)
+ show "XY.absolutely_continuous (joint_distribution X Y)"
+ proof (rule XY.absolutely_continuousI)
+ show "finite_measure_space XY.P (joint_distribution X Y)" by default
+ fix x assume "x \<in> space XY.P" and "XY.pair_measure {x} = 0"
+ then obtain a b where "(a, b) = x" and "a \<in> space S" "b \<in> space T"
+ and distr: "distribution X {a} * distribution Y {b} = 0"
+ by (cases x) (auto simp: pair_algebra_def)
+ with assms[THEN finite_random_variableD]
+ joint_distribution_Times_le_fst[of S X T Y "{a}" "{b}"]
+ joint_distribution_Times_le_snd[of S X T Y "{a}" "{b}"]
+ have "joint_distribution X Y {x} \<le> distribution Y {b}"
+ "joint_distribution X Y {x} \<le> distribution X {a}"
+ by auto
+ with distr show "joint_distribution X Y {x} = 0" by auto
+ qed
+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"
+ by (intro finite_random_variableD X Y mutual_information_commute_generic finite_variables_absolutely_continuous)
+
+lemma (in information_space) mutual_information_commute_simple:
+ assumes X: "simple_function X" and Y: "simple_function Y"
+ shows "\<I>(X;Y) = \<I>(Y;X)"
+ by (intro X Y simple_function_imp_finite_random_variable mutual_information_commute)
+
+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.
real (joint_distribution X Y {(x,y)}) *
log b (real (joint_distribution X Y {(x,y)}) /
(real (distribution X {x}) * real (distribution Y {y}))))"
- (is "?equality")
+ (is ?sum)
and mutual_information_positive_generic:
- "0 \<le> mutual_information b MX MY X Y" (is "?positive")
+ "0 \<le> mutual_information b MX MY X Y" (is ?positive)
proof -
- let ?P = "prod_measure_space MX MY"
- let ?\<mu> = "prod_measure MX (distribution X) MY (distribution Y)"
- let ?\<nu> = "joint_distribution X Y"
-
- interpret X: finite_prob_space MX "distribution X" by fact
- moreover interpret Y: finite_prob_space MY "distribution Y" by fact
- have ms_X: "measure_space MX (distribution X)"
- and ms_Y: "measure_space MY (distribution Y)"
- and fms: "finite_measure_space MX (distribution X)" "finite_measure_space MY (distribution Y)" by fact+
- have fms_P: "finite_measure_space ?P ?\<mu>"
- by (rule X.finite_measure_space_finite_prod_measure) fact
- then interpret P: finite_measure_space ?P ?\<mu> .
+ interpret X: finite_prob_space MX "distribution X" using MX by (rule distribution_finite_prob_space)
+ interpret Y: finite_prob_space MY "distribution Y" using MY by (rule distribution_finite_prob_space)
+ interpret XY: pair_finite_prob_space MX "distribution X" MY "distribution Y" by default
+ interpret P: finite_prob_space XY.P "joint_distribution X Y"
+ using assms by (intro joint_distribution_finite_prob_space)
- have fms_P': "finite_measure_space ?P ?\<nu>"
- 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 sigma_def)
- then interpret P': finite_measure_space ?P ?\<nu> .
-
- { fix x assume "x \<in> space ?P"
- hence in_MX: "{fst x} \<in> sets MX" "{snd x} \<in> sets MY" using X.sets_eq_Pow Y.sets_eq_Pow
- by (auto simp: prod_measure_space_def)
+ have P_ms: "finite_measure_space XY.P (joint_distribution X Y)" by default
+ have P_ps: "finite_prob_space XY.P (joint_distribution X Y)" by default
- assume "?\<mu> {x} = 0"
- with X.finite_prod_measure_times[OF fms(2), of "{fst x}" "{snd 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
-
- show ?equality
+ show ?sum
unfolding Let_def mutual_information_def
- using measure_0 fms_P fms_P' MX MY P.absolutely_continuous_def
- by (subst P.KL_divergence_eq_finite)
- (auto simp add: prod_measure_space_def prod_measure_times_finite
- finite_prob_space_eq setsum_cartesian_product' real_of_pinfreal_mult[symmetric])
+ by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
+ (auto simp add: pair_algebra_def setsum_cartesian_product' real_of_pinfreal_mult[symmetric])
show ?positive
- unfolding Let_def mutual_information_def using measure_0 b_gt_1
- proof (safe intro!: finite_prob_space.KL_divergence_positive_finite, simp_all)
- have "?\<mu> (space ?P) = 1"
- using X.top Y.top X.measure_space_1 Y.measure_space_1 fms
- by (simp add: prod_measure_space_def X.finite_prod_measure_times)
- with fms_P show "finite_prob_space ?P ?\<mu>"
- by (simp add: finite_prob_space_eq)
-
- from ms_X ms_Y X.top Y.top X.measure_space_1 Y.measure_space_1 Y.not_empty X_space Y_space
- have "?\<nu> (space ?P) = 1" unfolding measure_space_1[symmetric]
- by (auto intro!: arg_cong[where f="\<mu>"]
- simp add: prod_measure_space_def distribution_def vimage_Times comp_def)
- with fms_P' show "finite_prob_space ?P ?\<nu>"
- by (simp add: finite_prob_space_eq)
- qed
+ 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 finite_information_space) mutual_information_eq:
- "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
+lemma (in information_space) mutual_information_eq:
+ assumes "simple_function X" "simple_function Y"
+ shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) * log b (real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) /
(real (distribution X {x}) * real (distribution Y {y}))))"
- by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
+ using assms by (simp add: mutual_information_generic_eq)
-lemma (in finite_information_space) mutual_information_cong:
+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 "\<I>(X ; Y) = \<I>(X' ; Y')"
-proof -
- have "X ` space M = X' ` space M" using X by (auto intro!: image_eqI)
- moreover have "Y ` space M = Y' ` space M" using Y by (auto intro!: image_eqI)
- ultimately show ?thesis
- unfolding mutual_information_eq
- using
- assms[THEN distribution_cong]
- joint_distribution_cong[OF assms]
- by (auto intro!: setsum_cong)
-qed
+ 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 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)
+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 X" "simple_function Y"
+ shows "0 \<le> \<I>(X;Y)"
+ using assms by (simp add: mutual_information_positive_generic)
subsection {* Entropy *}
-definition (in prob_space)
- "entropy b s X = mutual_information b s s X X"
-
-abbreviation (in finite_information_space)
- finite_entropy ("\<H>'(_')") where
+abbreviation (in information_space)
+ entropy_Pow ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
-lemma (in finite_information_space) entropy_generic_eq:
- assumes MX: "finite_measure_space MX (distribution X)"
+lemma (in information_space) entropy_generic_eq:
+ assumes MX: "finite_random_variable MX X"
shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
proof -
+ interpret MX: finite_prob_space MX "distribution X" using MX by (rule distribution_finite_prob_space)
let "?X x" = "real (distribution X {x})"
let "?XX x y" = "real (joint_distribution X X {(x, y)})"
- interpret MX: finite_measure_space MX "distribution X" by fact
{ fix x y
have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
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)"
- unfolding distribution_def by (auto simp: mult_log_divide) }
+ unfolding distribution_def 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]
@@ -460,201 +419,327 @@
by (auto simp: setsum_cases MX.finite_space)
qed
-lemma (in finite_information_space) entropy_eq:
- "\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
- by (simp add: finite_measure_space entropy_generic_eq)
+lemma (in information_space) entropy_eq:
+ assumes "simple_function X"
+ shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
+ using assms by (simp add: entropy_generic_eq)
-lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
- unfolding entropy_def using mutual_information_positive .
+lemma (in information_space) entropy_positive:
+ "simple_function X \<Longrightarrow> 0 \<le> \<H>(X)"
+ unfolding entropy_def by (simp add: mutual_information_positive)
-lemma (in finite_information_space) entropy_certainty_eq_0:
- assumes "x \<in> X ` space M" and "distribution X {x} = 1"
+lemma (in information_space) entropy_certainty_eq_0:
+ assumes "simple_function X" and "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) \<rparr>" "distribution X"
- by (rule finite_prob_space_of_images)
-
+ using simple_function_imp_finite_random_variable[OF `simple_function X`]
+ by (rule distribution_finite_prob_space)
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 asm: "y \<noteq> x" "y \<in> X ` space M"
hence "{y} \<subseteq> X ` space M - {x}" by auto
from X.measure_mono[OF this] X0 asm
have "distribution X {y} = 0" by auto }
-
hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
using assms by auto
-
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 by (auto simp: y fi)
+ show ?thesis unfolding entropy_eq[OF `simple_function X`] by (auto simp: y fi)
qed
-lemma (in finite_information_space) entropy_le_card_not_0:
- "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
+lemma (in information_space) entropy_le_card_not_0:
+ assumes "simple_function X"
+ shows "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
proof -
let "?d x" = "distribution X {x}"
let "?p x" = "real (?d x)"
have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
- by (auto intro!: setsum_cong simp: entropy_eq setsum_negf[symmetric])
+ by (auto intro!: setsum_cong simp: entropy_eq[OF `simple_function X`] setsum_negf[symmetric] log_simps not_less)
also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
apply (rule log_setsum')
- using not_empty b_gt_1 finite_space sum_over_space_real_distribution
- by auto
+ using not_empty b_gt_1 `simple_function X` sum_over_space_real_distribution
+ by (auto simp: simple_function_def)
also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
- apply (rule arg_cong[where f="\<lambda>f. log b (\<Sum>x\<in>X`space M. f x)"])
- using distribution_finite[of X] by (auto simp: fun_eq_iff real_of_pinfreal_eq_0)
+ using distribution_finite[OF `simple_function X`[THEN simple_function_imp_random_variable], simplified]
+ by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) (auto simp: real_of_pinfreal_eq_0)
finally show ?thesis
- using finite_space by (auto simp: setsum_cases real_eq_of_nat)
+ using `simple_function X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
qed
-lemma (in finite_information_space) entropy_uniform_max:
+lemma (in information_space) entropy_uniform_max:
+ assumes "simple_function 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 -
- note uniform =
- finite_prob_space_of_images[of X, THEN finite_prob_space.uniform_prob, simplified]
-
+ interpret X: finite_prob_space "\<lparr> space = X ` space M, sets = Pow (X ` space M) \<rparr>" "distribution X"
+ using simple_function_imp_finite_random_variable[OF `simple_function X`]
+ by (rule distribution_finite_prob_space)
have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
- using finite_space not_empty by auto
-
+ using `simple_function X` not_empty by (auto simp: simple_function_def)
{ fix x assume "x \<in> X ` space M"
hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
- proof (rule uniform)
+ proof (rule X.uniform_prob[simplified])
fix x y assume "x \<in> X`space M" "y \<in> X`space M"
- from assms[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
+ from assms(2)[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
qed }
thus ?thesis
- using not_empty finite_space b_gt_1 card_gt0
- by (simp add: entropy_eq real_eq_of_nat[symmetric] log_divide)
+ using not_empty X.finite_space b_gt_1 card_gt0
+ by (simp add: entropy_eq[OF `simple_function X`] real_eq_of_nat[symmetric] log_simps)
qed
-lemma (in finite_information_space) entropy_le_card:
- "\<H>(X) \<le> log b (real (card (X ` space M)))"
+lemma (in information_space) entropy_le_card:
+ assumes "simple_function 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 finite_space not_empty unfolding card_gt_0_iff by auto
+ using `simple_function 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 unfolding entropy_eq by simp
+ 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 finite_space not_empty by (auto intro!: card_mono)
- note entropy_le_card_not_0
+ (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 finite_space not_empty `?A \<le> ?B`
- by (auto intro!: log_le simp: card_gt_0_iff)
+ 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 finite_information_space) entropy_commute:
- "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
+lemma (in information_space) entropy_commute:
+ assumes "simple_function X" "simple_function Y"
+ shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
proof -
+ have sf: "simple_function (\<lambda>x. (X x, Y x))" "simple_function (\<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 entropy_eq unfolding * setsum_reindex[OF inj]
+ unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
by (simp add: joint_distribution_commute[of Y X] split_beta)
qed
-lemma (in finite_information_space) entropy_eq_cartesian_sum:
- "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
+lemma (in information_space) entropy_eq_cartesian_product:
+ assumes "simple_function X" "simple_function Y"
+ shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
real (joint_distribution X Y {(x,y)}) *
log b (real (joint_distribution X Y {(x,y)})))"
proof -
+ have sf: "simple_function (\<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 finite_space
- unfolding entropy_eq neg_equal_iff_equal setsum_cartesian_product
- by (auto intro!: setsum_mono_zero_cong_left)
+ 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 M1 M2 M3 X Y Z \<equiv>
- mutual_information b M1 (prod_measure_space M2 M3) X (\<lambda>x. (Y x, Z x)) -
+ mutual_information b M1 (sigma (pair_algebra M2 M3)) X (\<lambda>x. (Y x, Z x)) -
mutual_information b M1 M3 X Z"
-abbreviation (in finite_information_space)
- finite_conditional_mutual_information ("\<I>'( _ ; _ | _ ')") where
+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) \<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"
-lemma (in finite_information_space) conditional_mutual_information_generic_eq:
- assumes MX: "finite_measure_space MX (distribution X)"
- assumes MY: "finite_measure_space MY (distribution Y)"
- assumes MZ: "finite_measure_space MZ (distribution 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.
- real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) *
- log b (real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) /
- (real (distribution X {x}) * real (joint_distribution Y Z {(y, z)})))) -
- (\<Sum>(x, y)\<in>space MX \<times> space MZ.
- real (joint_distribution X Z {(x, y)}) *
- log b (real (joint_distribution X Z {(x, y)}) / (real (distribution X {x}) * real (distribution Z {y}))))"
- using assms finite_measure_space_prod[OF MY MZ]
- unfolding conditional_mutual_information_def
- by (subst (1 2) mutual_information_generic_eq)
- (simp_all add: setsum_cartesian_product' finite_measure_space.finite_prod_measure_space)
-
-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.
+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.
real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
(real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
- by (subst conditional_mutual_information_generic_eq)
- (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
- finite_measure_space finite_product_prob_space_of_images sigma_def
- setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
- setsum_left_distrib[symmetric] setsum_real_distribution setsum_commute[where A="Y`space M"]
- real_of_pinfreal_mult[symmetric]
- cong: setsum_cong)
+ (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z)))")
+proof -
+ let ?YZ = "\<lambda>y z. real (joint_distribution Y Z {(y, z)})"
+ let ?X = "\<lambda>x. real (distribution X {x})"
+ let ?Z = "\<lambda>z. real (distribution Z {z})"
+
+ txt {* This proof is actually quiet easy, however we need to show that the
+ distributions are finite and the joint distributions are zero when one of
+ the variables distribution is also zero. *}
+
+ note finite_var = MX MY MZ
+ note random_var = finite_var[THEN finite_random_variableD]
+
+ note space_simps = space_pair_algebra space_sigma algebra.simps
+
+ note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
+ 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, simplified space_simps]
+ finite_distribution_order(5,6)[OF finite_var(1,3), simplified space_simps]
+
+ note finite = finite_var(1) YZ finite_var(3) XZ YZX
+ note finite[THEN finite_distribution_finite, simplified space_simps, simp]
+
+ 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 simp: space_simps)
-lemma (in finite_information_space) conditional_mutual_information_eq_mutual_information:
- "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
+ have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y 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"
+ then have *: "?XYZ x y z / (?XZ x z * ?YZdZ y z) =
+ (?XYZ x y z / (?X x * ?YZ y z)) / (?XZ x z / (?X x * ?Z z))"
+ using order1(3)
+ by (auto simp: real_of_pinfreal_mult[symmetric] real_of_pinfreal_eq_0)
+ show "?L x y z = ?R x y z"
+ proof cases
+ assume "?XYZ x y z \<noteq> 0"
+ with space b_gt_1 order1 order2 show ?thesis unfolding *
+ by (subst log_divide)
+ (auto simp: zero_less_divide_iff zero_less_real_of_pinfreal
+ real_of_pinfreal_eq_0 zero_less_mult_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_real_joint_distribution_singleton[OF finite_var(2) ZX, unfolded space_simps]
+ by (intro setsum_cong refl) simp
+ 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_algebra setsum_cartesian_product')
+ finally show ?thesis by simp
+qed
+
+lemma (in information_space) conditional_mutual_information_eq:
+ assumes "simple_function X" "simple_function Y" "simple_function Z"
+ shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
+ real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
+ log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
+ (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
+ using conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
+ by simp
+
+lemma (in information_space) conditional_mutual_information_eq_mutual_information:
+ assumes X: "simple_function X" and Y: "simple_function 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 (\<lambda>x. ())" by auto
show ?thesis
- unfolding conditional_mutual_information_eq mutual_information_eq
+ 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 finite_information_space) conditional_mutual_information_positive:
- "0 \<le> \<I>(X ; Y | Z)"
-proof -
+lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
+ unfolding distribution_def using measure_space_1 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:
+ assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. real (distribution X {a})) = 1"
+ using setsum_real_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 information_space) conditional_mutual_information_generic_positive:
+ assumes "finite_random_variable MX X" and "finite_random_variable MY Y" and "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 A" = "real (distribution (\<lambda>x. (X x, Y x, Z x)) A)"
let "?dXZ A" = "real (joint_distribution X Z A)"
let "?dYZ A" = "real (joint_distribution Y Z A)"
let "?dX A" = "real (distribution X A)"
let "?dZ A" = "real (distribution Z A)"
- let ?M = "X ` space M \<times> Y ` space M \<times> Z ` space M"
+ let ?M = "space MX \<times> space MY \<times> space MZ"
have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: fun_eq_iff)
- 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})"
+ note space_simps = space_pair_algebra space_sigma algebra.simps
+
+ note finite_var = assms
+ note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
+ note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
+ note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
+ note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
+ note XYZ = finite_random_variable_pairI[OF finite_var(1) YZ]
+ note finite = finite_var(3) YZ XZ XYZ
+ note finite = finite[THEN finite_distribution_finite, simplified space_simps]
+
+ 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 finite_var(2) ZX]
+ by (auto simp: space_simps)
+
+ note order = order
+ finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
+ finite_distribution_order(5,6)[OF finite_var(2,3), simplified space_simps]
+
+ 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 (intro setsum_cong) (auto intro!: arg_cong[where f="log b"] simp: real_of_pinfreal_mult[symmetric])
+ 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 not_empty by simp
+ 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_algebra
+ setsum_real_joint_distribution_singleton[OF `finite_random_variable MX X` YZ]
+ setsum_real_joint_distribution_singleton[OF `finite_random_variable MY Y` finite_var(3)]
+ setsum_real_distribution[OF `finite_random_variable MZ Z`])
+
fix x assume "x \<in> ?M"
let ?x = "(fst x, fst (snd x), snd (snd x))"
@@ -663,120 +748,180 @@
by (simp add: real_pinfreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
assume *: "0 < ?dXYZ {?x}"
- thus "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
- apply (rule_tac divide_pos_pos mult_pos_pos)+
- by (auto simp add: real_distribution_gt_0 intro: distribution_order(6) distribution_mono_gt_0)
- qed (simp_all add: setsum_cartesian_product' sum_over_space_real_distribution setsum_real_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})"
+ with `x \<in> ?M` show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+ using finite order
+ by (cases x)
+ (auto simp add: zero_less_real_of_pinfreal zero_less_mult_iff zero_less_divide_iff)
+ 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_real_distribution
+ by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
+ setsum_real_joint_distribution_singleton[OF finite_var(1,3)]
+ setsum_real_joint_distribution_singleton[OF finite_var(2,3)]
intro!: setsum_cong)
- finally show ?thesis
- unfolding conditional_mutual_information_eq sum_over_space_real_distribution
- by (simp add: real_of_pinfreal_mult[symmetric])
+ also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
+ unfolding setsum_real_distribution[OF finite_var(3)] by simp
+ finally show ?thesis by simp
qed
+lemma (in information_space) conditional_mutual_information_positive:
+ assumes "simple_function X" and "simple_function Y" and "simple_function Z"
+ shows "0 \<le> \<I>(X;Y|Z)"
+ using conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]]
+ by simp
+
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 finite_information_space)
- finite_conditional_entropy ("\<H>'(_ | _')") where
+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) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> 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 information_space) conditional_entropy_positive:
+ "simple_function X \<Longrightarrow> simple_function Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
+ unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
-lemma (in finite_information_space) conditional_entropy_generic_eq:
- assumes MX: "finite_measure_space MX (distribution X)"
- assumes MY: "finite_measure_space MZ (distribution Z)"
+lemma (in measure_space) empty_measureI: "A = {} \<Longrightarrow> \<mu> A = 0" by simp
+
+lemma (in information_space) conditional_entropy_generic_eq:
+ 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.
real (joint_distribution X Z {(x, z)}) *
log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
- unfolding conditional_entropy_def using assms
- apply (simp add: conditional_mutual_information_generic_eq
- setsum_cartesian_product' setsum_commute[of _ "space MZ"]
- setsum_negf[symmetric] setsum_subtractf[symmetric])
-proof (safe intro!: setsum_cong, simp)
- fix z x assume "z \<in> space MZ" "x \<in> space MX"
- let "?XXZ x'" = "real (joint_distribution X (\<lambda>x. (X x, Z x)) {(x, x', z)})"
- let "?XZ x'" = "real (joint_distribution X Z {(x', z)})"
- let "?X" = "real (distribution X {x})"
- interpret MX: finite_measure_space MX "distribution X" by fact
- have *: "\<And>A. A = {} \<Longrightarrow> prob A = 0" by simp
- have XXZ: "\<And>x'. ?XXZ x' = (if x' = x then ?XZ x else 0)"
- by (auto simp: distribution_def intro!: arg_cong[where f=prob] *)
- have "(\<Sum>x'\<in>space MX. ?XXZ x' * log b (?XXZ x') - (?XXZ x' * log b ?X + ?XXZ x' * log b (?XZ x'))) =
- (\<Sum>x'\<in>{x}. ?XZ x' * log b (?XZ x') - (?XZ x' * log b ?X + ?XZ x' * log b (?XZ x')))"
- using `x \<in> space MX` MX.finite_space
- by (safe intro!: setsum_mono_zero_cong_right)
- (auto split: split_if_asm simp: XXZ)
- then show "(\<Sum>x'\<in>space MX. ?XXZ x' * log b (?XXZ x') - (?XXZ x' * log b ?X + ?XXZ x' * log b (?XZ x'))) +
- ?XZ x * log b ?X = 0" by simp
+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 (real (?XXZ x y z) / (real (?XZ x z) * real (?XZ y z / ?Z 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' :: 'b and z assume "x' \<noteq> x"
+ then have "?XXZ x x' z = 0"
+ by (auto simp: distribution_def intro!: arg_cong[where f=\<mu>] empty_measureI) }
+ note this[simp]
+ { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
+ then have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z)
+ = (\<Sum>x'\<in>space MX. if x = x' then real (?XZ x z) * ?f x x z else 0)"
+ by (auto intro!: setsum_cong)
+ also have "\<dots> = real (?XZ x z) * ?f x x z"
+ using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
+ also have "\<dots> = real (?XZ x z) * log b (real (?Z z) / real (?XZ x z))"
+ by (auto simp: real_of_pinfreal_mult[symmetric])
+ also have "\<dots> = - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))"
+ using assms[THEN finite_distribution_finite]
+ using finite_distribution_order(6)[OF MX MZ]
+ by (auto simp: log_simps field_simps zero_less_mult_iff zero_less_real_of_pinfreal real_of_pinfreal_eq_0)
+ finally have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z) =
+ - real (?XZ x z) * log b (real (?XZ x z) / real (?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"] * simp del: divide_pinfreal_def
+ intro!: setsum_cong)
qed
-lemma (in finite_information_space) conditional_entropy_eq:
- "\<H>(X | Z) =
+lemma (in information_space) conditional_entropy_eq:
+ assumes "simple_function X" "simple_function Z"
+ shows "\<H>(X | Z) =
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
real (joint_distribution X Z {(x, z)}) *
log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
- by (simp add: finite_measure_space conditional_entropy_generic_eq)
+ using conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
+ by simp
-lemma (in finite_information_space) conditional_entropy_eq_ce_with_hypothesis:
- "\<H>(X | Y) =
+lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
+ assumes X: "simple_function X" and Y: "simple_function Y"
+ shows "\<H>(X | Y) =
-(\<Sum>y\<in>Y`space M. real (distribution Y {y}) *
(\<Sum>x\<in>X`space M. real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}) *
log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}))))"
- unfolding conditional_entropy_eq neg_equal_iff_equal
- apply (simp add: setsum_commute[of _ "Y`space M"] setsum_cartesian_product' setsum_divide_distrib[symmetric])
- apply (safe intro!: setsum_cong)
- using real_distribution_order'[of Y _ X _]
- by (auto simp add: setsum_subtractf[of _ _ "X`space M"])
+ unfolding conditional_entropy_eq[OF assms]
+ using finite_distribution_finite[OF finite_random_variable_pairI[OF assms[THEN simple_function_imp_finite_random_variable]]]
+ using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
+ using finite_distribution_finite[OF Y[THEN simple_function_imp_finite_random_variable]]
+ by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib real_of_pinfreal_eq_0
+ intro!: setsum_cong)
-lemma (in finite_information_space) conditional_entropy_eq_cartesian_sum:
- "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
+lemma (in information_space) conditional_entropy_eq_cartesian_product:
+ assumes "simple_function X" "simple_function Y"
+ shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
real (joint_distribution X Y {(x,y)}) *
log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
-proof -
- { 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 finite_space
- unfolding conditional_entropy_eq neg_equal_iff_equal setsum_cartesian_product
- by (auto intro!: setsum_mono_zero_cong_left)
-qed
+ unfolding conditional_entropy_eq[OF assms]
+ by (auto intro!: setsum_cong simp: setsum_cartesian_product')
subsection {* Equalities *}
-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
- by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product'
- setsum_left_distrib[symmetric] setsum_addf setsum_real_distribution)
+lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
+ assumes X: "simple_function X" and Z: "simple_function Z"
+ shows "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
+proof -
+ let "?XZ x z" = "real (joint_distribution X Z {(x, z)})"
+ let "?Z z" = "real (distribution Z {z})"
+ let "?X x" = "real (distribution X {x})"
+ note fX = X[THEN simple_function_imp_finite_random_variable]
+ note fZ = Z[THEN simple_function_imp_finite_random_variable]
+ note fX[THEN finite_distribution_finite, simp] and fZ[THEN finite_distribution_finite, simp]
+ 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 real_of_pinfreal_mult[symmetric] zero_less_mult_iff
+ zero_less_real_of_pinfreal field_simps real_of_pinfreal_eq_0 abs_mult) }
+ note * = this
+ show ?thesis
+ unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
+ using setsum_real_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_real_distribution)
+qed
-lemma (in finite_information_space) conditional_entropy_less_eq_entropy:
- "\<H>(X | Z) \<le> \<H>(X)"
+lemma (in information_space) conditional_entropy_less_eq_entropy:
+ assumes X: "simple_function X" and Z: "simple_function 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 .
- with mutual_information_positive[of X Z] entropy_positive[of X]
+ 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 finite_information_space) entropy_chain_rule:
- "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
- unfolding entropy_eq[of X] entropy_eq_cartesian_sum conditional_entropy_eq_cartesian_sum
- unfolding setsum_commute[of _ "X`space M"] setsum_negf[symmetric] setsum_addf[symmetric]
- by (rule setsum_cong)
- (simp_all add: setsum_negf setsum_addf setsum_subtractf setsum_real_distribution
- setsum_left_distrib[symmetric] joint_distribution_commute[of X Y])
+lemma (in information_space) entropy_chain_rule:
+ assumes X: "simple_function X" and Y: "simple_function Y"
+ shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
+proof -
+ let "?XY x y" = "real (joint_distribution X Y {(x, y)})"
+ let "?Y y" = "real (distribution Y {y})"
+ let "?X x" = "real (distribution X {x})"
+ note fX = X[THEN simple_function_imp_finite_random_variable]
+ note fY = Y[THEN simple_function_imp_finite_random_variable]
+ note fX[THEN finite_distribution_finite, simp] and fY[THEN finite_distribution_finite, simp]
+ 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 real_of_pinfreal_mult[symmetric] zero_less_mult_iff
+ zero_less_real_of_pinfreal field_simps real_of_pinfreal_eq_0 abs_mult) }
+ note * = this
+ show ?thesis
+ using setsum_real_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 *}
@@ -893,15 +1038,26 @@
finally show ?thesis .
qed
-lemma (in finite_information_space) entropy_partition:
+lemma (in information_space) entropy_partition:
+ assumes sf: "simple_function X" "simple_function P"
assumes svi: "subvimage (space M) X P"
shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
proof -
+ let "?XP x p" = "real (joint_distribution X P {(x, p)})"
+ let "?X x" = "real (distribution X {x})"
+ let "?P p" = "real (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 fX[THEN finite_distribution_finite, simp] and fP[THEN finite_distribution_finite, simp]
+ note finite_distribution_order[OF fX fP, simp]
have "(\<Sum>x\<in>X ` space M. real (distribution X {x}) * log b (real (distribution X {x}))) =
(\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
real (joint_distribution X P {(x, y)}) * log b (real (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)
+ 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 "real (joint_distribution X P {(X x, P p)}) * log b (real (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)
@@ -920,26 +1076,41 @@
log b (real (joint_distribution X P {(X x, P p)}))"
by (auto simp: distribution_def)
qed
- thus ?thesis
- unfolding entropy_eq conditional_entropy_eq
+ moreover have "\<And>x y. real (joint_distribution X P {(x, y)}) *
+ log b (real (joint_distribution X P {(x, y)}) / real (distribution P {y})) =
+ real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})) -
+ real (joint_distribution X P {(x, y)}) * log b (real (distribution 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_real_joint_distribution_singleton[OF fX fP]
by (simp add: setsum_cartesian_product' setsum_subtractf setsum_real_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)
+corollary (in information_space) entropy_data_processing:
+ assumes X: "simple_function X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
+proof -
+ note X
+ moreover have fX: "simple_function (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 finite_information_space) entropy_of_inj:
- assumes "inj_on f (X`space M)"
+corollary (in information_space) entropy_of_inj:
+ assumes X: "simple_function 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 .
+ show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
next
+ have sf: "simple_function (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 assms])
+ 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 .
+ using entropy_data_processing[OF sf] .
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
qed