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author | hoelzl |

Thu, 09 Jun 2011 13:55:11 +0200 | |

changeset 43339 | 9ba256ad6781 |

parent 43338 | a150d16bf77c |

child 43340 | 60e181c4eae4 |

jensens inequality

src/HOL/Probability/Lebesgue_Integration.thy | file | annotate | diff | comparison | revisions | |

src/HOL/Probability/Probability_Measure.thy | file | annotate | diff | comparison | revisions |

--- 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)" + by (simp add: positive_integral_max_0) + + 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)" + by (simp add: positive_integral_max_0) + + 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 + by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X) + 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)