author hoelzl Thu, 09 Jun 2011 13:55:11 +0200 changeset 43339 9ba256ad6781 parent 43338 a150d16bf77c child 43340 60e181c4eae4
jensens inequality
```--- a/src/HOL/Probability/Lebesgue_Integration.thy	Thu Jun 09 11:50:16 2011 +0200
+++ b/src/HOL/Probability/Lebesgue_Integration.thy	Thu Jun 09 13:55:11 2011 +0200
@@ -1712,6 +1712,12 @@
shows "integral\<^isup>L N f = integral\<^isup>L M f"
by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)

+lemma (in measure_space) integrable_cong_measure:
+  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
+  shows "integrable N f \<longleftrightarrow> integrable M f"
+  using assms
+  by (simp add: positive_integral_cong_measure[OF assms] integrable_def measurable_def)
+
lemma (in measure_space) integral_cong_AE:
assumes cong: "AE x. f x = g x"
shows "integral\<^isup>L M f = integral\<^isup>L M g"
@@ -1722,6 +1728,18 @@
unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
qed

+lemma (in measure_space) integrable_cong_AE:
+  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+  assumes "AE x. f x = g x"
+  shows "integrable M f = integrable M g"
+proof -
+  have "(\<integral>\<^isup>+ x. extreal (f x) \<partial>M) = (\<integral>\<^isup>+ x. extreal (g x) \<partial>M)"
+    "(\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) = (\<integral>\<^isup>+ x. extreal (- g x) \<partial>M)"
+    using assms by (auto intro!: positive_integral_cong_AE)
+  with assms show ?thesis
+    by (auto simp: integrable_def)
+qed
+
lemma (in measure_space) integrable_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
@@ -1767,6 +1785,48 @@
by (auto simp: borel[THEN positive_integral_vimage[OF T]])
qed

+lemma (in measure_space) integral_translated_density:
+  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
+    and g: "g \<in> borel_measurable M"
+    and N: "space N = space M" "sets N = sets M"
+    and density: "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
+      (is "\<And>A. _ \<Longrightarrow> _ = ?d A")
+  shows "integral\<^isup>L N g = (\<integral> x. f x * g x \<partial>M)" (is ?integral)
+    and "integrable N g = integrable M (\<lambda>x. f x * g x)" (is ?integrable)
+proof -
+  from f have ms: "measure_space (M\<lparr>measure := ?d\<rparr>)"
+    by (intro measure_space_density[where u="\<lambda>x. extreal (f x)"]) auto
+
+  from ms density N have "(\<integral>\<^isup>+ x. g x \<partial>N) =  (\<integral>\<^isup>+ x. max 0 (extreal (g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
+    unfolding positive_integral_max_0
+    by (intro measure_space.positive_integral_cong_measure) auto
+  also have "\<dots> = (\<integral>\<^isup>+ x. extreal (f x) * max 0 (extreal (g x)) \<partial>M)"
+    using f g by (intro positive_integral_translated_density) auto
+  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (extreal (f x * g x)) \<partial>M)"
+    using f by (intro positive_integral_cong_AE)
+               (auto simp: extreal_max_0 zero_le_mult_iff split: split_max)
+  finally have pos: "(\<integral>\<^isup>+ x. g x \<partial>N) = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
+
+  from ms density N have "(\<integral>\<^isup>+ x. - (g x) \<partial>N) =  (\<integral>\<^isup>+ x. max 0 (extreal (- g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
+    unfolding positive_integral_max_0
+    by (intro measure_space.positive_integral_cong_measure) auto
+  also have "\<dots> = (\<integral>\<^isup>+ x. extreal (f x) * max 0 (extreal (- g x)) \<partial>M)"
+    using f g by (intro positive_integral_translated_density) auto
+  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (extreal (- f x * g x)) \<partial>M)"
+    using f by (intro positive_integral_cong_AE)
+               (auto simp: extreal_max_0 mult_le_0_iff split: split_max)
+  finally have neg: "(\<integral>\<^isup>+ x. - g x \<partial>N) = (\<integral>\<^isup>+ x. - (f x * g x) \<partial>M)"
+
+  have g_N: "g \<in> borel_measurable N"
+    using g N unfolding measurable_def by simp
+
+  show ?integral ?integrable
+    unfolding lebesgue_integral_def integrable_def
+    using pos neg f g g_N by auto
+qed
+
lemma (in measure_space) integral_minus[intro, simp]:
assumes "integrable M f"
shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
@@ -2221,9 +2281,14 @@
by (simp add: \<mu>'_def lebesgue_integral_def positive_integral_const_If)
qed

+lemma indicator_less[simp]:
+  "indicator A x \<le> (indicator B x::extreal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
+  by (simp add: indicator_def not_le)
+
lemma (in finite_measure) integral_less_AE:
assumes int: "integrable M X" "integrable M Y"
-  assumes gt: "AE x. X x < Y x" and neq0: "\<mu> (space M) \<noteq> 0"
+  assumes A: "\<mu> A \<noteq> 0" "A \<in> sets M" "AE x. x \<in> A \<longrightarrow> X x \<noteq> Y x"
+  assumes gt: "AE x. X x \<le> Y x"
shows "integral\<^isup>L M X < integral\<^isup>L M Y"
proof -
have "integral\<^isup>L M X \<le> integral\<^isup>L M Y"
@@ -2231,24 +2296,30 @@
moreover
have "integral\<^isup>L M X \<noteq> integral\<^isup>L M Y"
proof
-    from gt have "AE x. Y x - X x \<noteq> 0"
-      by auto
-    then have \<mu>: "\<mu> {x\<in>space M. Y x - X x \<noteq> 0} = \<mu> (space M)"
-      using int
-      by (intro AE_measure borel_measurable_neq) (auto simp add: integrable_def)
-
assume eq: "integral\<^isup>L M X = integral\<^isup>L M Y"
have "integral\<^isup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^isup>L M (\<lambda>x. Y x - X x)"
using gt by (intro integral_cong_AE) auto
also have "\<dots> = 0"
using eq int by simp
-    finally show False
-      using \<mu> int neq0
-      by (subst (asm) integral_0_iff) auto
+    finally have "\<mu> {x \<in> space M. Y x - X x \<noteq> 0} = 0"
+      using int by (simp add: integral_0_iff)
+    moreover
+    have "(\<integral>\<^isup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^isup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
+      using A by (intro positive_integral_mono_AE) auto
+    then have "\<mu> A \<le> \<mu> {x \<in> space M. Y x - X x \<noteq> 0}"
+      using int A by (simp add: integrable_def)
+    moreover note `\<mu> A \<noteq> 0` positive_measure[OF `A \<in> sets M`]
+    ultimately show False by auto
qed
ultimately show ?thesis by auto
qed

+lemma (in finite_measure) integral_less_AE_space:
+  assumes int: "integrable M X" "integrable M Y"
+  assumes gt: "AE x. X x < Y x" "\<mu> (space M) \<noteq> 0"
+  shows "integral\<^isup>L M X < integral\<^isup>L M Y"
+  using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
+
lemma (in measure_space) integral_dominated_convergence:
assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
and w: "integrable M w"```
```--- a/src/HOL/Probability/Probability_Measure.thy	Thu Jun 09 11:50:16 2011 +0200
+++ b/src/HOL/Probability/Probability_Measure.thy	Thu Jun 09 13:55:11 2011 +0200
@@ -28,6 +28,14 @@
abbreviation (in prob_space)
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"

+lemma (in prob_space) prob_space_cong:
+  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
+  shows "prob_space N"
+proof -
+  interpret N: measure_space N by (intro measure_space_cong assms)
+  show ?thesis by default (insert assms measure_space_1, simp)
+qed
+
lemma (in prob_space) distribution_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
shows "distribution X = distribution Y"
@@ -54,15 +62,30 @@
lemma (in prob_space) distribution_positive[simp, intro]:
"0 \<le> distribution X A" unfolding distribution_def by auto

+lemma (in prob_space) not_zero_less_distribution[simp]:
+  "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
+  using distribution_positive[of X A] by arith
+
lemma (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 prob_space) not_empty: "space M \<noteq> {}"
+  using prob_space empty_measure' by auto
+
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
unfolding measure_space_1[symmetric]
using sets_into_space
by (intro measure_mono) auto

+lemma (in prob_space) AE_I_eq_1:
+  assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
+  shows "AE x. P x"
+proof (rule AE_I)
+  show "\<mu> (space M - {x \<in> space M. P x}) = 0"
+    using assms measure_space_1 by (simp add: measure_compl)
+qed (insert assms, auto)
+
lemma (in prob_space) distribution_1:
"distribution X A \<le> 1"
unfolding distribution_def by simp
@@ -245,6 +268,146 @@
using finite_measure_eq[of "X -` A \<inter> space M"]
by (auto simp: measurable_sets distribution_def)

+lemma (in prob_space) expectation_less:
+  assumes [simp]: "integrable M X"
+  assumes gt: "\<forall>x\<in>space M. X x < b"
+  shows "expectation X < b"
+proof -
+  have "expectation X < expectation (\<lambda>x. b)"
+    using gt measure_space_1
+    by (intro integral_less_AE) auto
+  then show ?thesis using prob_space by simp
+qed
+
+lemma (in prob_space) expectation_greater:
+  assumes [simp]: "integrable M X"
+  assumes gt: "\<forall>x\<in>space M. a < X x"
+  shows "a < expectation X"
+proof -
+  have "expectation (\<lambda>x. a) < expectation X"
+    using gt measure_space_1
+    by (intro integral_less_AE) auto
+  then show ?thesis using prob_space by simp
+qed
+
+lemma convex_le_Inf_differential:
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex_on I f"
+  assumes "x \<in> interior I" "y \<in> I"
+  shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
+    (is "_ \<ge> _ + Inf (?F x) * (y - x)")
+proof -
+  show ?thesis
+  proof (cases rule: linorder_cases)
+    assume "x < y"
+    moreover
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e . note e = this
+    moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
+    ultimately have "x < t" "t < y" "t \<in> ball x e"
+      by (auto simp: mem_ball dist_real_def field_simps split: split_min)
+    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e .
+    moreover def K \<equiv> "x - e / 2"
+    with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def)
+    ultimately have "K \<in> I" "K < x" "x \<in> I"
+      using interior_subset[of I] `x \<in> interior I` by auto
+
+    have "Inf (?F x) \<le> (f x - f y) / (x - y)"
+    proof (rule Inf_lower2)
+      show "(f x - f t) / (x - t) \<in> ?F x"
+        using `t \<in> I` `x < t` by auto
+      show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+        using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
+    next
+      fix y assume "y \<in> ?F x"
+      with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
+      show "(f K - f x) / (K - x) \<le> y" by auto
+    qed
+    then show ?thesis
+      using `x < y` by (simp add: field_simps)
+  next
+    assume "y < x"
+    moreover
+    have "open (interior I)" by auto
+    from openE[OF this `x \<in> interior I`] guess e . note e = this
+    moreover def t \<equiv> "x + e / 2"
+    ultimately have "x < t" "t \<in> ball x e"
+      by (auto simp: mem_ball dist_real_def field_simps)
+    with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+    have "(f x - f y) / (x - y) \<le> Inf (?F x)"
+    proof (rule Inf_greatest)
+      have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
+        using `y < x` by (auto simp: field_simps)
+      also
+      fix z  assume "z \<in> ?F x"
+      with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
+      have "(f y - f x) / (y - x) \<le> z" by auto
+      finally show "(f x - f y) / (x - y) \<le> z" .
+    next
+      have "open (interior I)" by auto
+      from openE[OF this `x \<in> interior I`] guess e . note e = this
+      then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def)
+      with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
+      then show "?F x \<noteq> {}" by blast
+    qed
+    then show ?thesis
+      using `y < x` by (simp add: field_simps)
+  qed simp
+qed
+
+lemma (in prob_space) jensens_inequality:
+  fixes a b :: real
+  assumes X: "integrable M X" "X ` space M \<subseteq> I"
+  assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
+  assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
+  shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
+proof -
+  let "?F x" = "Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
+  from not_empty X(2) have "I \<noteq> {}" by auto
+
+  from I have "open I" by auto
+
+  note I
+  moreover
+  { assume "I \<subseteq> {a <..}"
+    with X have "a < expectation X"
+      by (intro expectation_greater) auto }
+  moreover
+  { assume "I \<subseteq> {..< b}"
+    with X have "expectation X < b"
+      by (intro expectation_less) auto }
+  ultimately have "expectation X \<in> I"
+    by (elim disjE)  (auto simp: subset_eq)
+  moreover
+  { fix y assume y: "y \<in> I"
+    with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
+      by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
+  ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
+    by simp
+  also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
+  proof (rule Sup_least)
+    show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
+      using `I \<noteq> {}` by auto
+  next
+    fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
+    then guess x .. note x = this
+    have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
+      using prob_space
+    also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
+      using `x \<in> I` `open I` X(2)
+      by (intro integral_mono integral_add integral_cmult integral_diff
+                lebesgue_integral_const X q convex_le_Inf_differential)
+         (auto simp: interior_open)
+    finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
+  qed
+  finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
+qed
+
lemma (in prob_space) distribution_eq_translated_integral:
assumes "random_variable S X" "A \<in> sets S"
shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)"
@@ -722,9 +885,6 @@
unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
by auto

-lemma (in prob_space) not_empty: "space M \<noteq> {}"
-  using prob_space empty_measure' by auto
-
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
using measure_space_1 sum_over_space by simp

@@ -829,7 +989,7 @@
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
+  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)```