--- a/src/HOL/Probability/Information.thy Thu Sep 02 17:28:00 2010 +0200
+++ b/src/HOL/Probability/Information.thy Thu Sep 02 19:51:53 2010 +0200
@@ -2,11 +2,53 @@
imports Probability_Space Product_Measure Convex Radon_Nikodym
begin
+lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
+ by (subst log_le_cancel_iff) auto
+
+lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
+ by (subst log_less_cancel_iff) auto
+
+lemma setsum_cartesian_product':
+ "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
+ unfolding setsum_cartesian_product by simp
+
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:
@@ -105,51 +147,19 @@
finally show ?thesis .
qed
-lemma (in finite_prob_space) sum_over_space_distrib:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def measure_space_1[symmetric] using finite_space
- by (subst measure_finitely_additive'')
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=\<mu>])
-
-lemma (in finite_prob_space) sum_over_space_real_distribution:
- "(\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst real_finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+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
section "Information theory"
-definition
- "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)))"
-
locale finite_information_space = finite_prob_space +
fixes b :: real assumes b_gt_1: "1 < b"
-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 measure_space_1[symmetric]
- by (auto intro!: arg_cong[where f="\<mu>"] 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")
@@ -165,41 +175,6 @@
thus ?mult and ?div by auto
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
-
-lemma (in finite_information_space) distribution_finite:
- "distribution X A \<noteq> \<omega>"
- using measure_finite[of "X -` A \<inter> space M"]
- unfolding distribution_def sets_eq_Pow by auto
-
-lemma (in finite_information_space) real_distribution_gt_0[simp]:
- "0 < real (distribution Y y) \<longleftrightarrow> 0 < distribution Y y"
- using assms by (auto intro!: real_pinfreal_pos distribution_finite)
-
-lemma real_distribution_mult_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y * distribution X x)"
- unfolding real_of_pinfreal_mult[symmetric]
- using assms by (auto intro!: mult_pos_pos)
-
-lemma real_distribution_divide_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y / distribution X x)"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
- using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
-
-lemma real_distribution_mult_inverse_pos_pos:
- assumes "0 < distribution Y y"
- and "0 < distribution X x"
- shows "0 < real (distribution Y y * inverse (distribution X x))"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
- using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
-
ML {*
(* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
@@ -252,31 +227,14 @@
end
-lemma (in finite_measure_space) absolutely_continuousI:
- assumes "finite_measure_space M \<nu>"
- assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
- shows "absolutely_continuous \<nu>"
-proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
- fix N assume "\<mu> N = 0" "N \<subseteq> space M"
-
- interpret v: finite_measure_space M \<nu> by fact
+subsection "Kullback$-$Leibler divergence"
- have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp
- also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})"
- proof (rule v.measure_finitely_additive''[symmetric])
- show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
- show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
- fix x assume "x \<in> N" thus "{x} \<in> sets M" using `N \<subseteq> space M` sets_eq_Pow by auto
- qed
- also have "\<dots> = 0"
- proof (safe intro!: setsum_0')
- fix x assume "x \<in> N"
- hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
- hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
- thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
- qed
- finally show "\<nu> N = 0" .
-qed
+text {* The Kullback$-$Leibler divergence is also known as relative entropy or
+Kullback$-$Leibler distance. *}
+
+definition
+ "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 finite_measure_space) KL_divergence_eq_finite:
assumes v: "finite_measure_space M \<nu>"
@@ -285,19 +243,13 @@
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])
qed
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
- "(\<Sum>x\<in>space M. real (\<mu> {x})) = 1"
- using sum_over_space_eq_1 finite_measure by (simp add: real_of_pinfreal_setsum sets_eq_Pow)
-
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"
@@ -322,13 +274,15 @@
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 measure_finite[of "{x}"] sets_eq_Pow x
+ thus "0 < prob {x}" using finite_measure[of "{x}"] sets_eq_Pow x
by (cases "\<mu> {x}") simp_all }
qed auto
qed
thus "0 \<le> KL_divergence b M \<nu> \<mu>" using finite_sum_over_space_eq_1 by simp
qed
+subsection {* Mutual Information *}
+
definition (in prob_space)
"mutual_information b S T X Y =
KL_divergence b (prod_measure_space S T)
@@ -341,24 +295,6 @@
\<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 prod_measure_times_finite:
- assumes fms: "finite_measure_space M \<mu>" "finite_measure_space N \<nu>" and a: "a \<in> space M \<times> space N"
- shows "prod_measure M \<mu> N \<nu> {a} = \<mu> {fst a} * \<nu> {snd a}"
-proof (cases a)
- case (Pair b c)
- hence a_eq: "{a} = {b} \<times> {c}" by simp
-
- interpret M: finite_measure_space M \<mu> by fact
- interpret N: finite_measure_space N \<nu> by fact
-
- from finite_measure_space.finite_prod_measure_times[OF fms, of "{b}" "{c}"] M.sets_eq_Pow N.sets_eq_Pow a Pair
- show ?thesis unfolding a_eq by simp
-qed
-
-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
-
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)"
@@ -478,9 +414,26 @@
(real (distribution X {x}) * real (distribution Y {y}))))"
by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
+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 (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
+
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)
+subsection {* Entropy *}
+
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
@@ -488,32 +441,146 @@
finite_entropy ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> 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="\<mu>"])
+lemma (in finite_information_space) entropy_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
+proof -
+ 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) }
+ note remove_XX = this
+ show ?thesis
+ unfolding entropy_def mutual_information_generic_eq[OF MX MX]
+ unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
+ 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})))"
-proof -
- { fix f
- { fix x y
- have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
- hence "real (distribution (\<lambda>x. (X x, X x)) {(x,y)}) * f x y =
- (if x = y then real (distribution X {x}) * f x y else 0)"
- unfolding distribution_def by auto }
- hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. real (joint_distribution X X {(x, y)}) * f x y) =
- (\<Sum>x \<in> X ` space M. real (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
+ by (simp add: finite_measure_space entropy_generic_eq)
lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
unfolding entropy_def using mutual_information_positive .
+lemma (in finite_information_space) 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) \<rparr>" "distribution X"
+ 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
+ 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)
+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})))"
+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])
+ 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
+ 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: expand_fun_eq real_of_pinfreal_eq_0)
+ finally show ?thesis
+ using finite_space by (auto simp: setsum_cases real_eq_of_nat)
+qed
+
+lemma (in finite_information_space) 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>(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]
+
+ have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
+ using finite_space not_empty by auto
+
+ { fix x assume "x \<in> X ` space M"
+ hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
+ proof (rule uniform)
+ 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
+ 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)
+qed
+
+lemma (in finite_information_space) entropy_le_card:
+ "\<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
+ 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
+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
+ 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)
+ 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))"
+proof -
+ 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]
+ 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.
+ real (joint_distribution X Y {(x,y)}) *
+ log b (real (joint_distribution X Y {(x,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 entropy_eq neg_equal_iff_equal setsum_cartesian_product
+ by (auto intro!: setsum_mono_zero_cong_left)
+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)) -
@@ -527,87 +594,32 @@
\<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
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 (in finite_information_space) setsum_real_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. real (distribution Y {f x})) = real (distribution Z {c})"
- unfolding distribution_def assms
- using finite_space assms
- by (subst real_finite_measure_finite_Union[symmetric])
- (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_real_distribution:
- "(\<Sum>x \<in> X`space M. real (joint_distribution X Y {(x, y)})) = real (distribution Y {y})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X Y {(x, y)})) = real (distribution X {x})"
- "(\<Sum>x \<in> X`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution Y Z {(y, z)})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Z {(x, z)})"
- "(\<Sum>z \<in> Z`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Y {(x, y)})"
- by (auto intro!: inj_onI setsum_real_distribution_gen)
+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_sum:
- "\<I>(X ; Y | Z) =
- (\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` 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 (distribution (\<lambda>x. (Y x, Z x)) {(y, z)}))) -
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
- real (distribution (\<lambda>x. (X x, Z x)) {(x,z)}) * log b (real (distribution (\<lambda>x. (X x, Z x)) {(x,z)}) / real (distribution Z {z})))"
- (is "_ = ?rhs")
-proof -
- have setsum_product:
- "\<And>f x. (\<Sum>v\<in>(\<lambda>x. (Y x, Z x)) ` space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)}) * f v)
- = (\<Sum>v\<in>Y ` space M \<times> Z ` space M. real (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 "real (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
- by (subst mutual_information_eq_generic)
- (auto simp: prod_measure_space_def sigma_prod_sets_finite finite_space sigma_def
- 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_real_distribution
- cong: setsum_cong)
-qed
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.
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}))))"
- unfolding conditional_mutual_information_def Let_def mutual_information_eq
- by (subst mutual_information_eq_generic)
+ by (subst conditional_mutual_information_generic_eq)
(auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
- finite_prob_space_of_images finite_product_prob_space_of_images sigma_def
+ 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]
@@ -623,22 +635,6 @@
by (simp add: setsum_cartesian_product' distribution_remove_const)
qed
-lemma (in finite_prob_space) distribution_finite:
- "distribution X A \<noteq> \<omega>"
- by (auto simp: sets_eq_Pow distribution_def intro!: measure_finite)
-
-lemma (in finite_prob_space) real_distribution_order:
- shows "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution X {x})"
- and "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution Y {y})"
- and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution X {x})"
- and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution Y {y})"
- and "distribution X {x} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
- using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
- by auto
-
lemma (in finite_information_space) conditional_mutual_information_positive:
"0 \<le> \<I>(X ; Y | Z)"
proof -
@@ -682,6 +678,8 @@
by (simp add: real_of_pinfreal_mult[symmetric])
qed
+subsection {* Conditional Entropy *}
+
definition (in prob_space)
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
@@ -694,19 +692,69 @@
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_information_space) conditional_entropy_generic_eq:
+ assumes MX: "finite_measure_space MX (distribution X)"
+ assumes MY: "finite_measure_space MZ (distribution 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
+qed
+
lemma (in finite_information_space) conditional_entropy_eq:
"\<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)
+
+lemma (in finite_information_space) conditional_entropy_eq_ce_with_hypothesis:
+ "\<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"])
+
+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.
+ real (joint_distribution X Y {(x,y)}) *
+ log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
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 conditional_mutual_information_eq_sum
- conditional_entropy_def distribution_def *
- by (auto intro!: setsum_0')
+ { 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
+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
@@ -722,109 +770,15 @@
show ?thesis by auto
qed
-(* -------------Entropy of a RV with a certain event is zero---------------- *)
-
-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) \<rparr>" "distribution X"
- 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
- 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)
-qed
-(* --------------- upper bound on entropy for a rv ------------------------- *)
-
-lemma (in finite_prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def measure_space_1[symmetric]
- by (auto intro!: measure_mono simp: sets_eq_Pow)
-
-lemma (in finite_prob_space) real_distribution_1:
- "real (distribution X A) \<le> 1"
- unfolding real_pinfreal_1[symmetric]
- by (rule real_of_pinfreal_mono[OF _ distribution_1]) simp
+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 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 -
- 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])
- 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
- 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: expand_fun_eq real_of_pinfreal_eq_0)
- finally show ?thesis
- using finite_space by (auto simp: setsum_cases real_eq_of_nat)
-qed
-
-(* --------------- entropy is maximal for a uniform rv --------------------- *)
-
-lemma (in finite_prob_space) uniform_prob:
- assumes "x \<in> space M"
- assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
- shows "prob {x} = 1 / real (card (space M))"
-proof -
- have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
- using assms(2)[OF _ `x \<in> space M`] by blast
- have "1 = prob (space M)"
- using prob_space by auto
- also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
- using real_finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
- sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
- finite_space unfolding disjoint_family_on_def prob_space[symmetric]
- by (auto simp add:setsum_restrict_set)
- also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
- using prob_x by auto
- also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
- finally have one: "1 = real (card (space M)) * prob {x}"
- using real_eq_of_nat by auto
- hence two: "real (card (space M)) \<noteq> 0" by fastsimp
- from one have three: "prob {x} \<noteq> 0" by fastsimp
- thus ?thesis using one two three divide_cancel_right
- by (auto simp:field_simps)
-qed
-
-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>(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]
-
- have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
- using finite_space not_empty by auto
-
- { fix x assume "x \<in> X ` space M"
- hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
- proof (rule uniform)
- 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
- 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)
-qed
+section {* Partitioning *}
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
@@ -976,38 +930,6 @@
"\<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="\<mu>"])
-
-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="\<mu>"])
-
-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)"
--- a/src/HOL/Probability/Probability_Space.thy Thu Sep 02 17:28:00 2010 +0200
+++ b/src/HOL/Probability/Probability_Space.thy Thu Sep 02 19:51:53 2010 +0200
@@ -2,8 +2,6 @@
imports Lebesgue_Integration Radon_Nikodym
begin
-
-
locale prob_space = measure_space +
assumes measure_space_1: "\<mu> (space M) = 1"
@@ -33,6 +31,19 @@
abbreviation
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
+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="\<mu>"])
+
+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="\<mu>"])
+
lemma prob_space: "prob (space M) = 1"
unfolding measure_space_1 by simp
@@ -327,18 +338,158 @@
joint_distribution_restriction_snd[of X Y "{(x, y)}"]
by auto
-lemma (in finite_prob_space) finite_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
- by (simp add: finite_prob_space_eq finite_measure_space)
+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 finite_prob_space) 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 (in finite_prob_space) sum_over_space_distrib:
+ "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+ unfolding distribution_def measure_space_1[symmetric] using finite_space
+ by (subst measure_finitely_additive'')
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=\<mu>])
+
+lemma (in finite_prob_space) sum_over_space_real_distribution:
+ "(\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst real_finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+
+lemma (in finite_prob_space) finite_sum_over_space_eq_1:
+ "(\<Sum>x\<in>space M. real (\<mu> {x})) = 1"
+ using sum_over_space_eq_1 finite_measure by (simp add: real_of_pinfreal_setsum sets_eq_Pow)
+
+lemma (in finite_prob_space) distribution_finite:
+ "distribution X A \<noteq> \<omega>"
+ using finite_measure[of "X -` A \<inter> space M"]
+ unfolding distribution_def sets_eq_Pow by auto
+
+lemma (in finite_prob_space) real_distribution_gt_0[simp]:
+ "0 < real (distribution Y y) \<longleftrightarrow> 0 < distribution Y y"
+ using assms by (auto intro!: real_pinfreal_pos distribution_finite)
+
+lemma (in finite_prob_space) real_distribution_mult_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y * distribution X x)"
+ unfolding real_of_pinfreal_mult[symmetric]
+ using assms by (auto intro!: mult_pos_pos)
+
+lemma (in finite_prob_space) real_distribution_divide_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y / distribution X x)"
+ unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
+
+lemma (in finite_prob_space) real_distribution_mult_inverse_pos_pos:
+ assumes "0 < distribution Y y"
+ and "0 < distribution X x"
+ shows "0 < real (distribution Y y * inverse (distribution X x))"
+ unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
+
+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 measure_space_1[symmetric]
+ by (auto intro!: arg_cong[where f="\<mu>"] simp: distribution_def)
-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)
+lemma (in finite_prob_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_prob_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 (in finite_prob_space) setsum_real_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. real (distribution Y {f x})) = real (distribution Z {c})"
+ unfolding distribution_def assms
+ using finite_space assms
+ by (subst real_finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
+ intro!: arg_cong[where f=prob])
+
+lemma (in finite_prob_space) setsum_real_distribution:
+ "(\<Sum>x \<in> X`space M. real (joint_distribution X Y {(x, y)})) = real (distribution Y {y})"
+ "(\<Sum>y \<in> Y`space M. real (joint_distribution X Y {(x, y)})) = real (distribution X {x})"
+ "(\<Sum>x \<in> X`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution Y Z {(y, z)})"
+ "(\<Sum>y \<in> Y`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Z {(x, z)})"
+ "(\<Sum>z \<in> Z`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Y {(x, y)})"
+ by (auto intro!: inj_onI setsum_real_distribution_gen)
+
+lemma (in finite_prob_space) real_distribution_order:
+ shows "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution X {x})"
+ and "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution Y {y})"
+ and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution X {x})"
+ and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution Y {y})"
+ and "distribution X {x} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ and "distribution Y {y} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ by auto
+
+lemma (in prob_space) joint_distribution_remove[simp]:
+ "joint_distribution X X {(x, x)} = distribution X {x}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f="\<mu>"])
+
+lemma (in finite_prob_space) distribution_1:
+ "distribution X A \<le> 1"
+ unfolding distribution_def measure_space_1[symmetric]
+ by (auto intro!: measure_mono simp: sets_eq_Pow)
+
+lemma (in finite_prob_space) real_distribution_1:
+ "real (distribution X A) \<le> 1"
+ unfolding real_pinfreal_1[symmetric]
+ by (rule real_of_pinfreal_mono[OF _ distribution_1]) simp
+
+lemma (in finite_prob_space) uniform_prob:
+ assumes "x \<in> space M"
+ assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
+ shows "prob {x} = 1 / real (card (space M))"
+proof -
+ have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
+ using assms(2)[OF _ `x \<in> space M`] by blast
+ have "1 = prob (space M)"
+ using prob_space by auto
+ also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
+ using real_finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
+ sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
+ finite_space unfolding disjoint_family_on_def prob_space[symmetric]
+ by (auto simp add:setsum_restrict_set)
+ also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
+ using prob_x by auto
+ also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
+ finally have one: "1 = real (card (space M)) * prob {x}"
+ using real_eq_of_nat by auto
+ hence two: "real (card (space M)) \<noteq> 0" by fastsimp
+ from one have three: "prob {x} \<noteq> 0" by fastsimp
+ thus ?thesis using one two three divide_cancel_right
+ by (auto simp:field_simps)
qed
lemma (in prob_space) prob_space_subalgebra:
@@ -382,70 +533,54 @@
qed
lemma (in finite_prob_space) finite_measure_space:
+ fixes X :: "'a \<Rightarrow> 'x"
shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
(is "finite_measure_space ?S _")
proof (rule finite_measure_spaceI, simp_all)
show "finite (X ` space M)" using finite_space by simp
-
- show "positive (distribution X)"
- unfolding distribution_def positive_def using sets_eq_Pow by 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 = {}"
- hence "X -` x \<inter> space M \<inter> (X -` y \<inter> space M) = {}" by auto
- from additive[unfolded additive_def, rule_format, OF x y] this
- finite_measure[OF x] finite_measure[OF y]
- have "\<mu> (((X -` x) \<union> (X -` y)) \<inter> space M) =
- \<mu> ((X -` x) \<inter> space M) + \<mu> ((X -` y) \<inter> space M)"
- by (subst Int_Un_distrib2) auto
- thus "\<mu> ((X -` x \<union> X -` y) \<inter> space M) = \<mu> (X -` x \<inter> space M) + \<mu> (X -` y \<inter> space M)"
- by auto
- qed
-
- { fix x assume "x \<in> X ` space M" thus "distribution X {x} \<noteq> \<omega>"
- unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
+next
+ fix A B :: "'x set" assume "A \<inter> B = {}"
+ then show "distribution X (A \<union> B) = distribution X A + distribution X B"
+ unfolding distribution_def
+ by (subst measure_additive)
+ (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
qed
+lemma (in finite_prob_space) finite_prob_space_of_images:
+ "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M)\<rparr> (distribution X)"
+ by (simp add: finite_prob_space_eq finite_measure_space)
+
+lemma (in prob_space) joint_distribution_commute:
+ "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
+
+lemma (in finite_prob_space) real_distribution_order':
+ shows "real (distribution X {x}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ and "real (distribution Y {y}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ by auto
+
lemma (in finite_prob_space) finite_product_measure_space:
+ fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
assumes "finite s1" "finite s2"
shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2)\<rparr> (joint_distribution X Y)"
(is "finite_measure_space ?M ?D")
-proof (rule finite_Pow_additivity_sufficient)
- show "positive ?D"
- unfolding positive_def using assms sets_eq_Pow
- by (simp add: distribution_def)
-
- show "additive ?M ?D" unfolding additive_def
- proof safe
- fix x y
- have A: "((\<lambda>x. (X x, Y x)) -` x) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
- have B: "((\<lambda>x. (X x, Y x)) -` y) \<inter> space M \<in> sets M" using assms sets_eq_Pow by auto
- assume "x \<inter> y = {}"
- hence "(\<lambda>x. (X x, Y x)) -` x \<inter> space M \<inter> ((\<lambda>x. (X x, Y x)) -` y \<inter> space M) = {}"
- by auto
- from additive[unfolded additive_def, rule_format, OF A B] this
- finite_measure[OF A] finite_measure[OF B]
- show "?D (x \<union> y) = ?D x + ?D y"
- apply (simp add: distribution_def)
- apply (subst Int_Un_distrib2)
- by (auto simp: real_of_pinfreal_add)
- qed
-
- show "finite (space ?M)"
+proof (rule finite_measure_spaceI, simp_all)
+ show "finite (s1 \<times> s2)"
using assms by auto
-
- show "sets ?M = Pow (space ?M)"
- by simp
-
- { fix x assume "x \<in> space ?M" thus "?D {x} \<noteq> \<omega>"
- unfolding distribution_def by (auto intro!: finite_measure simp: sets_eq_Pow) }
+ show "joint_distribution X Y (s1\<times>s2) \<noteq> \<omega>"
+ using distribution_finite .
+next
+ fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
+ then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
+ unfolding distribution_def
+ by (subst measure_additive)
+ (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
qed
-lemma (in finite_measure_space) finite_product_measure_space_of_images:
+lemma (in finite_prob_space) finite_product_measure_space_of_images:
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
sets = Pow (X ` space M \<times> Y ` space M) \<rparr>
(joint_distribution X Y)"