author | immler |
Tue, 27 Nov 2012 11:29:47 +0100 | |
changeset 50244 | de72bbe42190 |
parent 50123 | 69b35a75caf3 |
child 53015 | a1119cf551e8 |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Independent_Family.thy |
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Add formalization of probabilistic independence for families of sets
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Author: Johannes Hölzl, TU München |
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Add formalization of probabilistic independence for families of sets
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*) |
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Add formalization of probabilistic independence for families of sets
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|
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Add formalization of probabilistic independence for families of sets
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header {* Independent families of events, event sets, and random variables *} |
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|
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Add formalization of probabilistic independence for families of sets
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theory Independent_Family |
47694 | 8 |
imports Probability_Measure Infinite_Product_Measure |
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begin |
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10 |
|
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definition (in prob_space) |
42983 | 12 |
"indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and> |
42981 | 13 |
(\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))))" |
14 |
||
15 |
definition (in prob_space) |
|
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"indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV" |
|
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|
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definition (in prob_space) |
49784
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indep_events_def_alt: "indep_events A I \<longleftrightarrow> indep_sets (\<lambda>i. {A i}) I" |
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|
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lemma (in prob_space) indep_events_def: |
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"indep_events A I \<longleftrightarrow> (A`I \<subseteq> events) \<and> |
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(\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))" |
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unfolding indep_events_def_alt indep_sets_def |
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apply (simp add: Ball_def Pi_iff image_subset_iff_funcset) |
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apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong) |
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apply auto |
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done |
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29 |
|
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definition (in prob_space) |
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"indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV" |
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|
47694 | 33 |
lemma (in prob_space) indep_sets_cong: |
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"I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> indep_sets G J" |
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by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+ |
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||
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lemma (in prob_space) indep_events_finite_index_events: |
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"indep_events F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)" |
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by (auto simp: indep_events_def) |
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||
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lemma (in prob_space) indep_sets_finite_index_sets: |
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"indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)" |
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proof (intro iffI allI impI) |
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assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J" |
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show "indep_sets F I" unfolding indep_sets_def |
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proof (intro conjI ballI allI impI) |
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fix i assume "i \<in> I" |
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with *[THEN spec, of "{i}"] show "F i \<subseteq> events" |
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by (auto simp: indep_sets_def) |
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qed (insert *, auto simp: indep_sets_def) |
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qed (auto simp: indep_sets_def) |
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52 |
|
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lemma (in prob_space) indep_sets_mono_index: |
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"J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J" |
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unfolding indep_sets_def by auto |
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|
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lemma (in prob_space) indep_sets_mono_sets: |
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assumes indep: "indep_sets F I" |
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assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i" |
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shows "indep_sets G I" |
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proof - |
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have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)" |
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using mono by auto |
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moreover have "\<And>A J. J \<subseteq> I \<Longrightarrow> A \<in> (\<Pi> j\<in>J. G j) \<Longrightarrow> A \<in> (\<Pi> j\<in>J. F j)" |
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using mono by (auto simp: Pi_iff) |
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ultimately show ?thesis |
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using indep by (auto simp: indep_sets_def) |
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qed |
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|
49772 | 70 |
lemma (in prob_space) indep_sets_mono: |
71 |
assumes indep: "indep_sets F I" |
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assumes mono: "J \<subseteq> I" "\<And>i. i\<in>J \<Longrightarrow> G i \<subseteq> F i" |
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shows "indep_sets G J" |
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apply (rule indep_sets_mono_sets) |
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apply (rule indep_sets_mono_index) |
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apply (fact +) |
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done |
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||
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lemma (in prob_space) indep_setsI: |
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assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" |
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and "\<And>A J. J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> (\<forall>j\<in>J. A j \<in> F j) \<Longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
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shows "indep_sets F I" |
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using assms unfolding indep_sets_def by (auto simp: Pi_iff) |
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84 |
|
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lemma (in prob_space) indep_setsD: |
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assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j" |
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shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
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88 |
using assms unfolding indep_sets_def by auto |
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89 |
|
42982 | 90 |
lemma (in prob_space) indep_setI: |
91 |
assumes ev: "A \<subseteq> events" "B \<subseteq> events" |
|
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and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b" |
|
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shows "indep_set A B" |
|
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unfolding indep_set_def |
|
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proof (rule indep_setsI) |
|
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fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV" |
|
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and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)" |
|
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have "J \<in> Pow UNIV" by auto |
|
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with F `J \<noteq> {}` indep[of "F True" "F False"] |
|
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show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))" |
|
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unfolding UNIV_bool Pow_insert by (auto simp: ac_simps) |
|
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qed (auto split: bool.split simp: ev) |
|
103 |
||
104 |
lemma (in prob_space) indep_setD: |
|
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assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B" |
|
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shows "prob (a \<inter> b) = prob a * prob b" |
|
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using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev |
|
108 |
by (simp add: ac_simps UNIV_bool) |
|
109 |
||
110 |
lemma (in prob_space) |
|
111 |
assumes indep: "indep_set A B" |
|
42983 | 112 |
shows indep_setD_ev1: "A \<subseteq> events" |
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and indep_setD_ev2: "B \<subseteq> events" |
|
42982 | 114 |
using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto |
115 |
||
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116 |
lemma (in prob_space) indep_sets_dynkin: |
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117 |
assumes indep: "indep_sets F I" |
47694 | 118 |
shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I" |
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119 |
(is "indep_sets ?F I") |
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120 |
proof (subst indep_sets_finite_index_sets, intro allI impI ballI) |
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121 |
fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}" |
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122 |
with indep have "indep_sets F J" |
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123 |
by (subst (asm) indep_sets_finite_index_sets) auto |
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124 |
{ fix J K assume "indep_sets F K" |
46731 | 125 |
let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i" |
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126 |
assume "finite J" "J \<subseteq> K" |
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127 |
then have "indep_sets (?G J) K" |
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128 |
proof induct |
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129 |
case (insert j J) |
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130 |
moreover def G \<equiv> "?G J" |
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131 |
ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K" |
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132 |
by (auto simp: indep_sets_def) |
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133 |
let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }" |
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134 |
{ fix X assume X: "X \<in> events" |
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135 |
assume indep: "\<And>J A. J \<noteq> {} \<Longrightarrow> J \<subseteq> K \<Longrightarrow> finite J \<Longrightarrow> j \<notin> J \<Longrightarrow> (\<forall>i\<in>J. A i \<in> G i) |
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136 |
\<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))" |
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137 |
have "indep_sets (G(j := {X})) K" |
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138 |
proof (rule indep_setsI) |
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139 |
fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events" |
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|
140 |
using G X by auto |
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|
141 |
next |
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142 |
fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i" |
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hoelzl
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143 |
show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
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|
144 |
proof cases |
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|
145 |
assume "j \<in> J" |
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|
146 |
with J have "A j = X" by auto |
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hoelzl
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|
147 |
show ?thesis |
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|
148 |
proof cases |
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|
149 |
assume "J = {j}" then show ?thesis by simp |
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|
150 |
next |
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151 |
assume "J \<noteq> {j}" |
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hoelzl
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|
152 |
have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)" |
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hoelzl
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|
153 |
using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm) |
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|
154 |
also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
155 |
proof (rule indep) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
156 |
show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
157 |
using J `J \<noteq> {j}` `j \<in> J` by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
158 |
show "\<forall>i\<in>J - {j}. A i \<in> G i" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
159 |
using J by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
160 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
161 |
also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
162 |
using `A j = X` by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
163 |
also have "\<dots> = (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
164 |
unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob (A i)"] |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
165 |
using `j \<in> J` by (simp add: insert_absorb) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
166 |
finally show ?thesis . |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
167 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
168 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
169 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
170 |
with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
171 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
172 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
173 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
174 |
qed } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
175 |
note indep_sets_insert = this |
47694 | 176 |
have "dynkin_system (space M) ?D" |
42987 | 177 |
proof (rule dynkin_systemI', simp_all cong del: indep_sets_cong, safe) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
178 |
show "indep_sets (G(j := {{}})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
179 |
by (rule indep_sets_insert) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
180 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
181 |
fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
182 |
show "indep_sets (G(j := {space M - X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
183 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
184 |
fix J A assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "j \<notin> J" and A: "\<forall>i\<in>J. A i \<in> G i" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
185 |
then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
186 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
187 |
have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
188 |
prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
189 |
using A_sets sets.sets_into_space[of _ M] X `J \<noteq> {}` |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
190 |
by (auto intro!: arg_cong[where f=prob] split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
191 |
also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
192 |
using J `J \<noteq> {}` `j \<notin> J` A_sets X sets.sets_into_space |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
193 |
by (auto intro!: finite_measure_Diff sets.finite_INT split: split_if_asm) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
194 |
finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
195 |
prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" . |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
196 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
197 |
have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
198 |
using J A `finite J` by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
199 |
then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
200 |
using prob_space by simp } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
201 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
202 |
have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
203 |
using J A `j \<in> K` by (intro indep_setsD[OF G']) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
204 |
then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
205 |
using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
206 |
ultimately have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = (prob (space M) - prob X) * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
207 |
by (simp add: field_simps) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
208 |
also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
209 |
using X A by (simp add: finite_measure_compl) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
210 |
finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" . |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
211 |
qed (insert X, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
212 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
213 |
fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
214 |
then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
215 |
show "indep_sets (G(j := {\<Union>k. F k})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
216 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
217 |
fix J A assume J: "j \<notin> J" "J \<noteq> {}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> G i" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
218 |
then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
219 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
220 |
have "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
221 |
using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
222 |
moreover have "(\<lambda>k. prob (\<Inter>i\<in>insert j J. (A(j := F k)) i)) sums prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
223 |
proof (rule finite_measure_UNION) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
224 |
show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
225 |
using disj by (rule disjoint_family_on_bisimulation) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
226 |
show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
227 |
using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: sets.Int) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
228 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
229 |
moreover { fix k |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
230 |
from J A `j \<in> K` have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
231 |
by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
232 |
also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
233 |
using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
234 |
finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
235 |
ultimately have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
236 |
by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
237 |
moreover |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
238 |
have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * prob (\<Inter>i\<in>J. A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
239 |
using disj F(1) by (intro finite_measure_UNION sums_mult2) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
240 |
then have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * (\<Prod>i\<in>J. prob (A i)))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
241 |
using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
242 |
ultimately |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
243 |
show "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. F k) * (\<Prod>j\<in>J. prob (A j))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
244 |
by (auto dest!: sums_unique) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
245 |
qed (insert F, auto) |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
246 |
qed (insert sets.sets_into_space, auto) |
47694 | 247 |
then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}" |
248 |
proof (rule dynkin_system.dynkin_subset, safe) |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
249 |
fix X assume "X \<in> G j" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
250 |
then show "X \<in> events" using G `j \<in> K` by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
251 |
from `indep_sets G K` |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
252 |
show "indep_sets (G(j := {X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
253 |
by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
254 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
255 |
have "indep_sets (G(j:=?D)) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
256 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
257 |
fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
258 |
using G(2) by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
259 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
260 |
fix A J assume J: "J\<noteq>{}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> (G(j := ?D)) i" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
261 |
show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
262 |
proof cases |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
263 |
assume "j \<in> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
264 |
with A have indep: "indep_sets (G(j := {A j})) K" by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
265 |
from J A show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
266 |
by (intro indep_setsD[OF indep]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
267 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
268 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
269 |
with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
270 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
271 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
272 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
273 |
qed |
47694 | 274 |
then have "indep_sets (G(j := dynkin (space M) (G j))) K" |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
275 |
by (rule indep_sets_mono_sets) (insert mono, auto) |
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changeset
|
276 |
then show ?case |
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diff
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|
277 |
by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def) |
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parents:
diff
changeset
|
278 |
qed (insert `indep_sets F K`, simp) } |
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parents:
diff
changeset
|
279 |
from this[OF `indep_sets F J` `finite J` subset_refl] |
47694 | 280 |
show "indep_sets ?F J" |
42861
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diff
changeset
|
281 |
by (rule indep_sets_mono_sets) auto |
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hoelzl
parents:
diff
changeset
|
282 |
qed |
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hoelzl
parents:
diff
changeset
|
283 |
|
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diff
changeset
|
284 |
lemma (in prob_space) indep_sets_sigma: |
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parents:
diff
changeset
|
285 |
assumes indep: "indep_sets F I" |
47694 | 286 |
assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" |
287 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
|
42861
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parents:
diff
changeset
|
288 |
proof - |
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parents:
diff
changeset
|
289 |
from indep_sets_dynkin[OF indep] |
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diff
changeset
|
290 |
show ?thesis |
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parents:
diff
changeset
|
291 |
proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable) |
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parents:
diff
changeset
|
292 |
fix i assume "i \<in> I" |
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|
293 |
with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def) |
50244
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immler
parents:
50123
diff
changeset
|
294 |
with sets.sets_into_space show "F i \<subseteq> Pow (space M)" by auto |
42861
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diff
changeset
|
295 |
qed |
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parents:
diff
changeset
|
296 |
qed |
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diff
changeset
|
297 |
|
42987 | 298 |
lemma (in prob_space) indep_sets_sigma_sets_iff: |
47694 | 299 |
assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" |
42987 | 300 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I" |
301 |
proof |
|
302 |
assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
|
47694 | 303 |
by (rule indep_sets_sigma) fact |
42987 | 304 |
next |
305 |
assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I" |
|
306 |
by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic) |
|
307 |
qed |
|
308 |
||
49794 | 309 |
definition (in prob_space) |
310 |
indep_vars_def2: "indep_vars M' X I \<longleftrightarrow> |
|
49781 | 311 |
(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> |
312 |
indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" |
|
49794 | 313 |
|
314 |
definition (in prob_space) |
|
315 |
"indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV" |
|
316 |
||
317 |
lemma (in prob_space) indep_vars_def: |
|
318 |
"indep_vars M' X I \<longleftrightarrow> |
|
319 |
(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> |
|
320 |
indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" |
|
321 |
unfolding indep_vars_def2 |
|
49781 | 322 |
apply (rule conj_cong[OF refl]) |
49794 | 323 |
apply (rule indep_sets_sigma_sets_iff[symmetric]) |
49781 | 324 |
apply (auto simp: Int_stable_def) |
325 |
apply (rule_tac x="A \<inter> Aa" in exI) |
|
326 |
apply auto |
|
327 |
done |
|
328 |
||
49794 | 329 |
lemma (in prob_space) indep_var_eq: |
330 |
"indep_var S X T Y \<longleftrightarrow> |
|
331 |
(random_variable S X \<and> random_variable T Y) \<and> |
|
332 |
indep_set |
|
333 |
(sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S}) |
|
334 |
(sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})" |
|
335 |
unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool |
|
336 |
by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext) |
|
337 |
(auto split: bool.split) |
|
338 |
||
42861
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parents:
diff
changeset
|
339 |
lemma (in prob_space) indep_sets2_eq: |
42981 | 340 |
"indep_set A B \<longleftrightarrow> A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)" |
341 |
unfolding indep_set_def |
|
42861
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parents:
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changeset
|
342 |
proof (intro iffI ballI conjI) |
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parents:
diff
changeset
|
343 |
assume indep: "indep_sets (bool_case A B) UNIV" |
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parents:
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changeset
|
344 |
{ fix a b assume "a \<in> A" "b \<in> B" |
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hoelzl
parents:
diff
changeset
|
345 |
with indep_setsD[OF indep, of UNIV "bool_case a b"] |
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parents:
diff
changeset
|
346 |
show "prob (a \<inter> b) = prob a * prob b" |
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Add formalization of probabilistic independence for families of sets
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parents:
diff
changeset
|
347 |
unfolding UNIV_bool by (simp add: ac_simps) } |
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hoelzl
parents:
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changeset
|
348 |
from indep show "A \<subseteq> events" "B \<subseteq> events" |
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parents:
diff
changeset
|
349 |
unfolding indep_sets_def UNIV_bool by auto |
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hoelzl
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diff
changeset
|
350 |
next |
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parents:
diff
changeset
|
351 |
assume *: "A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)" |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
352 |
show "indep_sets (bool_case A B) UNIV" |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
353 |
proof (rule indep_setsI) |
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Add formalization of probabilistic independence for families of sets
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parents:
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changeset
|
354 |
fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
355 |
using * by (auto split: bool.split) |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
356 |
next |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
357 |
fix J X assume "J \<noteq> {}" "J \<subseteq> UNIV" and X: "\<forall>j\<in>J. X j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)" |
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hoelzl
parents:
diff
changeset
|
358 |
then have "J = {True} \<or> J = {False} \<or> J = {True,False}" |
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Add formalization of probabilistic independence for families of sets
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parents:
diff
changeset
|
359 |
by (auto simp: UNIV_bool) |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
360 |
then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))" |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
361 |
using X * by auto |
16375b493b64
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hoelzl
parents:
diff
changeset
|
362 |
qed |
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hoelzl
parents:
diff
changeset
|
363 |
qed |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
364 |
|
42981 | 365 |
lemma (in prob_space) indep_set_sigma_sets: |
366 |
assumes "indep_set A B" |
|
47694 | 367 |
assumes A: "Int_stable A" and B: "Int_stable B" |
42981 | 368 |
shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)" |
42861
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hoelzl
parents:
diff
changeset
|
369 |
proof - |
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hoelzl
parents:
diff
changeset
|
370 |
have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV" |
47694 | 371 |
proof (rule indep_sets_sigma) |
42861
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hoelzl
parents:
diff
changeset
|
372 |
show "indep_sets (bool_case A B) UNIV" |
42981 | 373 |
by (rule `indep_set A B`[unfolded indep_set_def]) |
47694 | 374 |
fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)" |
42861
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
375 |
using A B by (cases i) auto |
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hoelzl
parents:
diff
changeset
|
376 |
qed |
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hoelzl
parents:
diff
changeset
|
377 |
then show ?thesis |
42981 | 378 |
unfolding indep_set_def |
42861
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
379 |
by (rule indep_sets_mono_sets) (auto split: bool.split) |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
380 |
qed |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
381 |
|
42981 | 382 |
lemma (in prob_space) indep_sets_collect_sigma: |
383 |
fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set" |
|
384 |
assumes indep: "indep_sets E (\<Union>j\<in>J. I j)" |
|
47694 | 385 |
assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)" |
42981 | 386 |
assumes disjoint: "disjoint_family_on I J" |
387 |
shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J" |
|
388 |
proof - |
|
46731 | 389 |
let ?E = "\<lambda>j. {\<Inter>k\<in>K. E' k| E' K. finite K \<and> K \<noteq> {} \<and> K \<subseteq> I j \<and> (\<forall>k\<in>K. E' k \<in> E k) }" |
42981 | 390 |
|
42983 | 391 |
from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events" |
42981 | 392 |
unfolding indep_sets_def by auto |
393 |
{ fix j |
|
47694 | 394 |
let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)" |
42981 | 395 |
assume "j \<in> J" |
47694 | 396 |
from E[OF this] interpret S: sigma_algebra "space M" ?S |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
397 |
using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto |
42981 | 398 |
|
399 |
have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)" |
|
400 |
proof (rule sigma_sets_eqI) |
|
401 |
fix A assume "A \<in> (\<Union>i\<in>I j. E i)" |
|
402 |
then guess i .. |
|
403 |
then show "A \<in> sigma_sets (space M) (?E j)" |
|
47694 | 404 |
by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"]) |
42981 | 405 |
next |
406 |
fix A assume "A \<in> ?E j" |
|
407 |
then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k" |
|
408 |
and A: "A = (\<Inter>k\<in>K. E' k)" |
|
409 |
by auto |
|
47694 | 410 |
then have "A \<in> ?S" unfolding A |
411 |
by (safe intro!: S.finite_INT) auto |
|
42981 | 412 |
then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)" |
47694 | 413 |
by simp |
42981 | 414 |
qed } |
415 |
moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J" |
|
47694 | 416 |
proof (rule indep_sets_sigma) |
42981 | 417 |
show "indep_sets ?E J" |
418 |
proof (intro indep_setsI) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
419 |
fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: sets.finite_INT) |
42981 | 420 |
next |
421 |
fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K" |
|
422 |
and "\<forall>j\<in>K. A j \<in> ?E j" |
|
423 |
then have "\<forall>j\<in>K. \<exists>E' L. A j = (\<Inter>l\<in>L. E' l) \<and> finite L \<and> L \<noteq> {} \<and> L \<subseteq> I j \<and> (\<forall>l\<in>L. E' l \<in> E l)" |
|
424 |
by simp |
|
425 |
from bchoice[OF this] guess E' .. |
|
426 |
from bchoice[OF this] obtain L |
|
427 |
where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)" |
|
428 |
and L: "\<And>j. j\<in>K \<Longrightarrow> finite (L j)" "\<And>j. j\<in>K \<Longrightarrow> L j \<noteq> {}" "\<And>j. j\<in>K \<Longrightarrow> L j \<subseteq> I j" |
|
429 |
and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l" |
|
430 |
by auto |
|
431 |
||
432 |
{ fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k" |
|
433 |
have "k = j" |
|
434 |
proof (rule ccontr) |
|
435 |
assume "k \<noteq> j" |
|
436 |
with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}" |
|
437 |
unfolding disjoint_family_on_def by auto |
|
438 |
with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`] |
|
439 |
show False using `l \<in> L k` `l \<in> L j` by auto |
|
440 |
qed } |
|
441 |
note L_inj = this |
|
442 |
||
443 |
def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)" |
|
444 |
{ fix x j l assume *: "j \<in> K" "l \<in> L j" |
|
445 |
have "k l = j" unfolding k_def |
|
446 |
proof (rule some_equality) |
|
447 |
fix k assume "k \<in> K \<and> l \<in> L k" |
|
448 |
with * L_inj show "k = j" by auto |
|
449 |
qed (insert *, simp) } |
|
450 |
note k_simp[simp] = this |
|
46731 | 451 |
let ?E' = "\<lambda>l. E' (k l) l" |
42981 | 452 |
have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)" |
453 |
by (auto simp: A intro!: arg_cong[where f=prob]) |
|
454 |
also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))" |
|
455 |
using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono) |
|
456 |
also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))" |
|
457 |
using K L L_inj by (subst setprod_UN_disjoint) auto |
|
458 |
also have "\<dots> = (\<Prod>j\<in>K. prob (A j))" |
|
459 |
using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast |
|
460 |
finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" . |
|
461 |
qed |
|
462 |
next |
|
463 |
fix j assume "j \<in> J" |
|
47694 | 464 |
show "Int_stable (?E j)" |
42981 | 465 |
proof (rule Int_stableI) |
466 |
fix a assume "a \<in> ?E j" then obtain Ka Ea |
|
467 |
where a: "a = (\<Inter>k\<in>Ka. Ea k)" "finite Ka" "Ka \<noteq> {}" "Ka \<subseteq> I j" "\<And>k. k\<in>Ka \<Longrightarrow> Ea k \<in> E k" by auto |
|
468 |
fix b assume "b \<in> ?E j" then obtain Kb Eb |
|
469 |
where b: "b = (\<Inter>k\<in>Kb. Eb k)" "finite Kb" "Kb \<noteq> {}" "Kb \<subseteq> I j" "\<And>k. k\<in>Kb \<Longrightarrow> Eb k \<in> E k" by auto |
|
470 |
let ?A = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})" |
|
471 |
have "a \<inter> b = INTER (Ka \<union> Kb) ?A" |
|
472 |
by (simp add: a b set_eq_iff) auto |
|
473 |
with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j" |
|
474 |
by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto |
|
475 |
qed |
|
476 |
qed |
|
477 |
ultimately show ?thesis |
|
478 |
by (simp cong: indep_sets_cong) |
|
479 |
qed |
|
480 |
||
49772 | 481 |
definition (in prob_space) tail_events where |
482 |
"tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
42982 | 483 |
|
49772 | 484 |
lemma (in prob_space) tail_events_sets: |
485 |
assumes A: "\<And>i::nat. A i \<subseteq> events" |
|
486 |
shows "tail_events A \<subseteq> events" |
|
487 |
proof |
|
488 |
fix X assume X: "X \<in> tail_events A" |
|
42982 | 489 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
49772 | 490 |
from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def) |
42982 | 491 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
42983 | 492 |
then show "X \<in> events" |
42982 | 493 |
by induct (insert A, auto) |
494 |
qed |
|
495 |
||
49772 | 496 |
lemma (in prob_space) sigma_algebra_tail_events: |
47694 | 497 |
assumes "\<And>i::nat. sigma_algebra (space M) (A i)" |
49772 | 498 |
shows "sigma_algebra (space M) (tail_events A)" |
499 |
unfolding tail_events_def |
|
42982 | 500 |
proof (simp add: sigma_algebra_iff2, safe) |
501 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
47694 | 502 |
interpret A: sigma_algebra "space M" "A i" for i by fact |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
503 |
{ fix X x assume "X \<in> ?A" "x \<in> X" |
42982 | 504 |
then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto |
505 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
|
506 |
then have "X \<subseteq> space M" |
|
507 |
by induct (insert A.sets_into_space, auto) |
|
508 |
with `x \<in> X` show "x \<in> space M" by auto } |
|
509 |
{ fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A" |
|
510 |
then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)" |
|
511 |
by (intro sigma_sets.Union) auto } |
|
512 |
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty) |
|
513 |
||
514 |
lemma (in prob_space) kolmogorov_0_1_law: |
|
515 |
fixes A :: "nat \<Rightarrow> 'a set set" |
|
47694 | 516 |
assumes "\<And>i::nat. sigma_algebra (space M) (A i)" |
42982 | 517 |
assumes indep: "indep_sets A UNIV" |
49772 | 518 |
and X: "X \<in> tail_events A" |
42982 | 519 |
shows "prob X = 0 \<or> prob X = 1" |
520 |
proof - |
|
49781 | 521 |
have A: "\<And>i. A i \<subseteq> events" |
522 |
using indep unfolding indep_sets_def by simp |
|
523 |
||
47694 | 524 |
let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}" |
525 |
interpret A: sigma_algebra "space M" "A i" for i by fact |
|
49772 | 526 |
interpret T: sigma_algebra "space M" "tail_events A" |
527 |
by (rule sigma_algebra_tail_events) fact |
|
42982 | 528 |
have "X \<subseteq> space M" using T.space_closed X by auto |
529 |
||
42983 | 530 |
have X_in: "X \<in> events" |
49772 | 531 |
using tail_events_sets A X by auto |
42982 | 532 |
|
47694 | 533 |
interpret D: dynkin_system "space M" ?D |
42982 | 534 |
proof (rule dynkin_systemI) |
47694 | 535 |
fix D assume "D \<in> ?D" then show "D \<subseteq> space M" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
536 |
using sets.sets_into_space by auto |
42982 | 537 |
next |
47694 | 538 |
show "space M \<in> ?D" |
42982 | 539 |
using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2) |
540 |
next |
|
47694 | 541 |
fix A assume A: "A \<in> ?D" |
42982 | 542 |
have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))" |
543 |
using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob]) |
|
544 |
also have "\<dots> = prob X - prob (X \<inter> A)" |
|
545 |
using X_in A by (intro finite_measure_Diff) auto |
|
546 |
also have "\<dots> = prob X * prob (space M) - prob X * prob A" |
|
547 |
using A prob_space by auto |
|
548 |
also have "\<dots> = prob X * prob (space M - A)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
549 |
using X_in A sets.sets_into_space |
42982 | 550 |
by (subst finite_measure_Diff) (auto simp: field_simps) |
47694 | 551 |
finally show "space M - A \<in> ?D" |
42982 | 552 |
using A `X \<subseteq> space M` by auto |
553 |
next |
|
47694 | 554 |
fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D" |
42982 | 555 |
then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)" |
556 |
by auto |
|
557 |
have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)" |
|
558 |
proof (rule finite_measure_UNION) |
|
559 |
show "range (\<lambda>i. X \<inter> F i) \<subseteq> events" |
|
560 |
using F X_in by auto |
|
561 |
show "disjoint_family (\<lambda>i. X \<inter> F i)" |
|
562 |
using dis by (rule disjoint_family_on_bisimulation) auto |
|
563 |
qed |
|
564 |
with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))" |
|
565 |
by simp |
|
566 |
moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
43920
diff
changeset
|
567 |
by (intro sums_mult finite_measure_UNION F dis) |
42982 | 568 |
ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)" |
569 |
by (auto dest!: sums_unique) |
|
47694 | 570 |
with F show "(\<Union>i. F i) \<in> ?D" |
42982 | 571 |
by auto |
572 |
qed |
|
573 |
||
574 |
{ fix n |
|
575 |
have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV" |
|
576 |
proof (rule indep_sets_collect_sigma) |
|
577 |
have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _") |
|
578 |
by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq) |
|
579 |
with indep show "indep_sets A ?U" by simp |
|
580 |
show "disjoint_family (bool_case {..n} {Suc n..})" |
|
581 |
unfolding disjoint_family_on_def by (auto split: bool.split) |
|
582 |
fix m |
|
47694 | 583 |
show "Int_stable (A m)" |
42982 | 584 |
unfolding Int_stable_def using A.Int by auto |
585 |
qed |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
586 |
also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) = |
42982 | 587 |
bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))" |
588 |
by (auto intro!: ext split: bool.split) |
|
589 |
finally have indep: "indep_set (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))" |
|
590 |
unfolding indep_set_def by simp |
|
591 |
||
47694 | 592 |
have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D" |
42982 | 593 |
proof (simp add: subset_eq, rule) |
594 |
fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
595 |
have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)" |
|
49772 | 596 |
using X unfolding tail_events_def by simp |
42982 | 597 |
from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D |
598 |
show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D" |
|
599 |
by (auto simp add: ac_simps) |
|
600 |
qed } |
|
47694 | 601 |
then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _") |
42982 | 602 |
by auto |
603 |
||
49772 | 604 |
note `X \<in> tail_events A` |
47694 | 605 |
also { |
606 |
have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A" |
|
607 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
49772 | 608 |
then have "tail_events A \<subseteq> sigma_sets (space M) ?A" |
609 |
unfolding tail_events_def by auto } |
|
47694 | 610 |
also have "sigma_sets (space M) ?A = dynkin (space M) ?A" |
42982 | 611 |
proof (rule sigma_eq_dynkin) |
612 |
{ fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
613 |
then have "B \<subseteq> space M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
614 |
by induct (insert A sets.sets_into_space[of _ M], auto) } |
47694 | 615 |
then show "?A \<subseteq> Pow (space M)" by auto |
616 |
show "Int_stable ?A" |
|
42982 | 617 |
proof (rule Int_stableI) |
618 |
fix a assume "a \<in> ?A" then guess n .. note a = this |
|
619 |
fix b assume "b \<in> ?A" then guess m .. note b = this |
|
47694 | 620 |
interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
621 |
using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto |
42982 | 622 |
have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
623 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
624 |
with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
625 |
moreover |
|
626 |
have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
627 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
628 |
with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
629 |
ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
47694 | 630 |
using Amn.Int[of a b] by simp |
42982 | 631 |
then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto |
632 |
qed |
|
633 |
qed |
|
47694 | 634 |
also have "dynkin (space M) ?A \<subseteq> ?D" |
635 |
using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset) |
|
636 |
finally show ?thesis by auto |
|
42982 | 637 |
qed |
638 |
||
42985 | 639 |
lemma (in prob_space) borel_0_1_law: |
640 |
fixes F :: "nat \<Rightarrow> 'a set" |
|
49781 | 641 |
assumes F2: "indep_events F UNIV" |
42985 | 642 |
shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1" |
643 |
proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"]) |
|
49781 | 644 |
have F1: "range F \<subseteq> events" |
645 |
using F2 by (simp add: indep_events_def subset_eq) |
|
47694 | 646 |
{ fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
647 |
using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space |
47694 | 648 |
by auto } |
42985 | 649 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV" |
47694 | 650 |
proof (rule indep_sets_sigma) |
42985 | 651 |
show "indep_sets (\<lambda>i. {F i}) UNIV" |
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49781
diff
changeset
|
652 |
unfolding indep_events_def_alt[symmetric] by fact |
47694 | 653 |
fix i show "Int_stable {F i}" |
42985 | 654 |
unfolding Int_stable_def by simp |
655 |
qed |
|
46731 | 656 |
let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i" |
49772 | 657 |
show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})" |
658 |
unfolding tail_events_def |
|
42985 | 659 |
proof |
660 |
fix j |
|
47694 | 661 |
interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
662 |
using order_trans[OF F1 sets.space_closed] |
47694 | 663 |
by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq) |
42985 | 664 |
have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)" |
665 |
by (intro decseq_SucI INT_decseq_offset UN_mono) auto |
|
47694 | 666 |
also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
667 |
using order_trans[OF F1 sets.space_closed] |
42985 | 668 |
by (safe intro!: S.countable_INT S.countable_UN) |
47694 | 669 |
(auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI) |
42985 | 670 |
finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
47694 | 671 |
by simp |
42985 | 672 |
qed |
673 |
qed |
|
674 |
||
42987 | 675 |
lemma (in prob_space) indep_sets_finite: |
676 |
assumes I: "I \<noteq> {}" "finite I" |
|
677 |
and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" |
|
678 |
shows "indep_sets F I \<longleftrightarrow> (\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j)))" |
|
679 |
proof |
|
680 |
assume *: "indep_sets F I" |
|
681 |
from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
682 |
by (intro indep_setsD[OF *] ballI) auto |
|
683 |
next |
|
684 |
assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
685 |
show "indep_sets F I" |
|
686 |
proof (rule indep_setsI[OF F(1)]) |
|
687 |
fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" |
|
688 |
assume A: "\<forall>j\<in>J. A j \<in> F j" |
|
46731 | 689 |
let ?A = "\<lambda>j. if j \<in> J then A j else space M" |
42987 | 690 |
have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
691 |
using subset_trans[OF F(1) sets.space_closed] J A |
42987 | 692 |
by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast |
693 |
also |
|
694 |
from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _") |
|
695 |
by (auto split: split_if_asm) |
|
696 |
with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))" |
|
697 |
by auto |
|
698 |
also have "\<dots> = (\<Prod>j\<in>J. prob (A j))" |
|
699 |
unfolding if_distrib setprod.If_cases[OF `finite I`] |
|
700 |
using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1) |
|
701 |
finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" .. |
|
702 |
qed |
|
703 |
qed |
|
704 |
||
42989 | 705 |
lemma (in prob_space) indep_vars_finite: |
42987 | 706 |
fixes I :: "'i set" |
707 |
assumes I: "I \<noteq> {}" "finite I" |
|
47694 | 708 |
and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)" |
709 |
and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)" |
|
710 |
and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)" |
|
711 |
and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> E i" and closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M' i))" |
|
712 |
shows "indep_vars M' X I \<longleftrightarrow> |
|
713 |
(\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))" |
|
42987 | 714 |
proof - |
715 |
from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)" |
|
716 |
unfolding measurable_def by simp |
|
717 |
||
718 |
{ fix i assume "i\<in>I" |
|
47694 | 719 |
from closed[OF `i \<in> I`] |
720 |
have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} |
|
721 |
= sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}" |
|
722 |
unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`] |
|
42987 | 723 |
by (subst sigma_sets_sigma_sets_eq) auto } |
47694 | 724 |
note sigma_sets_X = this |
42987 | 725 |
|
726 |
{ fix i assume "i\<in>I" |
|
47694 | 727 |
have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}" |
42987 | 728 |
proof (rule Int_stableI) |
47694 | 729 |
fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
730 |
then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto |
|
42987 | 731 |
moreover |
47694 | 732 |
fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
733 |
then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto |
|
42987 | 734 |
moreover |
735 |
have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto |
|
736 |
moreover note Int_stable[OF `i \<in> I`] |
|
737 |
ultimately |
|
47694 | 738 |
show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
42987 | 739 |
by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD) |
740 |
qed } |
|
47694 | 741 |
note indep_sets_X = indep_sets_sigma_sets_iff[OF this] |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
742 |
|
42987 | 743 |
{ fix i assume "i \<in> I" |
47694 | 744 |
{ fix A assume "A \<in> E i" |
745 |
with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto |
|
42987 | 746 |
moreover |
47694 | 747 |
from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto |
42987 | 748 |
ultimately |
749 |
have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) } |
|
750 |
with X[OF `i\<in>I`] space[OF `i\<in>I`] |
|
47694 | 751 |
have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events" |
752 |
"space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
|
42987 | 753 |
by (auto intro!: exI[of _ "space (M' i)"]) } |
47694 | 754 |
note indep_sets_finite_X = indep_sets_finite[OF I this] |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
755 |
|
47694 | 756 |
have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) = |
757 |
(\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))" |
|
42987 | 758 |
(is "?L = ?R") |
759 |
proof safe |
|
47694 | 760 |
fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)" |
42987 | 761 |
from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}` |
762 |
show "prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M))" |
|
763 |
by (auto simp add: Pi_iff) |
|
764 |
next |
|
47694 | 765 |
fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})" |
766 |
from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto |
|
42987 | 767 |
from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M" |
47694 | 768 |
"B \<in> (\<Pi> i\<in>I. E i)" by auto |
42987 | 769 |
from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}` |
770 |
show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))" |
|
771 |
by simp |
|
772 |
qed |
|
773 |
then show ?thesis using `I \<noteq> {}` |
|
47694 | 774 |
by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong) |
42988 | 775 |
qed |
776 |
||
42989 | 777 |
lemma (in prob_space) indep_vars_compose: |
778 |
assumes "indep_vars M' X I" |
|
47694 | 779 |
assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)" |
42989 | 780 |
shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I" |
781 |
unfolding indep_vars_def |
|
42988 | 782 |
proof |
42989 | 783 |
from rv `indep_vars M' X I` |
42988 | 784 |
show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)" |
47694 | 785 |
by (auto simp: indep_vars_def) |
42988 | 786 |
|
787 |
have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
42989 | 788 |
using `indep_vars M' X I` by (simp add: indep_vars_def) |
42988 | 789 |
then show "indep_sets (\<lambda>i. sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)}) I" |
790 |
proof (rule indep_sets_mono_sets) |
|
791 |
fix i assume "i \<in> I" |
|
42989 | 792 |
with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)" |
793 |
unfolding indep_vars_def measurable_def by auto |
|
42988 | 794 |
{ fix A assume "A \<in> sets (N i)" |
795 |
then have "\<exists>B. (Y i \<circ> X i) -` A \<inter> space M = X i -` B \<inter> space M \<and> B \<in> sets (M' i)" |
|
796 |
by (intro exI[of _ "Y i -` A \<inter> space (M' i)"]) |
|
797 |
(auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) } |
|
798 |
then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq> |
|
799 |
sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
800 |
by (intro sigma_sets_subseteq) (auto simp: vimage_compose) |
|
801 |
qed |
|
802 |
qed |
|
803 |
||
47694 | 804 |
lemma (in prob_space) indep_varsD_finite: |
42989 | 805 |
assumes X: "indep_vars M' X I" |
42988 | 806 |
assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)" |
807 |
shows "prob (\<Inter>i\<in>I. X i -` A i \<inter> space M) = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))" |
|
808 |
proof (rule indep_setsD) |
|
809 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
42989 | 810 |
using X by (auto simp: indep_vars_def) |
42988 | 811 |
show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto |
812 |
show "\<forall>i\<in>I. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
47694 | 813 |
using I by auto |
42988 | 814 |
qed |
815 |
||
47694 | 816 |
lemma (in prob_space) indep_varsD: |
817 |
assumes X: "indep_vars M' X I" |
|
818 |
assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)" |
|
819 |
shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))" |
|
820 |
proof (rule indep_setsD) |
|
821 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
822 |
using X by (auto simp: indep_vars_def) |
|
823 |
show "\<forall>i\<in>J. X i -` A i \<inter> space M \<in> sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
824 |
using I by auto |
|
825 |
qed fact+ |
|
826 |
||
827 |
lemma (in prob_space) indep_vars_iff_distr_eq_PiM: |
|
828 |
fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b" |
|
829 |
assumes "I \<noteq> {}" |
|
42988 | 830 |
assumes rv: "\<And>i. random_variable (M' i) (X i)" |
42989 | 831 |
shows "indep_vars M' X I \<longleftrightarrow> |
47694 | 832 |
distr M (\<Pi>\<^isub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i))" |
42988 | 833 |
proof - |
47694 | 834 |
let ?P = "\<Pi>\<^isub>M i\<in>I. M' i" |
835 |
let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x" |
|
836 |
let ?D = "distr M ?P ?X" |
|
837 |
have X: "random_variable ?P ?X" by (intro measurable_restrict rv) |
|
838 |
interpret D: prob_space ?D by (intro prob_space_distr X) |
|
42988 | 839 |
|
47694 | 840 |
let ?D' = "\<lambda>i. distr M (M' i) (X i)" |
841 |
let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)" |
|
842 |
interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv) |
|
843 |
interpret P: product_prob_space ?D' I .. |
|
844 |
||
42988 | 845 |
show ?thesis |
47694 | 846 |
proof |
42989 | 847 |
assume "indep_vars M' X I" |
47694 | 848 |
show "?D = ?P'" |
849 |
proof (rule measure_eqI_generator_eq) |
|
850 |
show "Int_stable (prod_algebra I M')" |
|
851 |
by (rule Int_stable_prod_algebra) |
|
852 |
show "prod_algebra I M' \<subseteq> Pow (space ?P)" |
|
853 |
using prod_algebra_sets_into_space by (simp add: space_PiM) |
|
854 |
show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')" |
|
855 |
by (simp add: sets_PiM space_PiM) |
|
856 |
show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')" |
|
857 |
by (simp add: sets_PiM space_PiM cong: prod_algebra_cong) |
|
858 |
let ?A = "\<lambda>i. \<Pi>\<^isub>E i\<in>I. space (M' i)" |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49781
diff
changeset
|
859 |
show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')" |
47694 | 860 |
by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong) |
861 |
{ fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto } |
|
862 |
next |
|
863 |
fix E assume E: "E \<in> prod_algebra I M'" |
|
864 |
from prod_algebraE[OF E] guess J Y . note J = this |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
865 |
|
47694 | 866 |
from E have "E \<in> sets ?P" by (auto simp: sets_PiM) |
867 |
then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)" |
|
868 |
by (simp add: emeasure_distr X) |
|
869 |
also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)" |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
870 |
using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm) |
47694 | 871 |
also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))" |
872 |
using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J] |
|
873 |
by (auto simp: emeasure_eq_measure setprod_ereal) |
|
874 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" |
|
875 |
using rv J by (simp add: emeasure_distr) |
|
876 |
also have "\<dots> = emeasure ?P' E" |
|
877 |
using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def) |
|
878 |
finally show "emeasure ?D E = emeasure ?P' E" . |
|
42988 | 879 |
qed |
880 |
next |
|
47694 | 881 |
assume "?D = ?P'" |
882 |
show "indep_vars M' X I" unfolding indep_vars_def |
|
883 |
proof (intro conjI indep_setsI ballI rv) |
|
884 |
fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
885 |
by (auto intro!: sets.sigma_sets_subset measurable_sets rv) |
42988 | 886 |
next |
47694 | 887 |
fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" |
888 |
assume Y': "\<forall>j\<in>J. Y' j \<in> sigma_sets (space M) {X j -` A \<inter> space M |A. A \<in> sets (M' j)}" |
|
889 |
have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" |
|
42988 | 890 |
proof |
47694 | 891 |
fix j assume "j \<in> J" |
892 |
from Y'[rule_format, OF this] rv[of j] |
|
893 |
show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" |
|
894 |
by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"]) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
895 |
(auto dest: measurable_space simp: sets.sigma_sets_eq) |
42988 | 896 |
qed |
47694 | 897 |
from bchoice[OF this] obtain Y where |
898 |
Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto |
|
899 |
let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)" |
|
900 |
from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M" |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
901 |
using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm) |
47694 | 902 |
then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)" |
903 |
by simp |
|
904 |
also have "\<dots> = emeasure ?D ?E" |
|
905 |
using Y J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto |
|
906 |
also have "\<dots> = emeasure ?P' ?E" |
|
907 |
using `?D = ?P'` by simp |
|
908 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" |
|
909 |
using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def) |
|
910 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))" |
|
911 |
using rv J Y by (simp add: emeasure_distr) |
|
912 |
finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" . |
|
913 |
then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))" |
|
914 |
by (auto simp: emeasure_eq_measure setprod_ereal) |
|
42988 | 915 |
qed |
916 |
qed |
|
42987 | 917 |
qed |
918 |
||
42989 | 919 |
lemma (in prob_space) indep_varD: |
920 |
assumes indep: "indep_var Ma A Mb B" |
|
921 |
assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb" |
|
922 |
shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = |
|
923 |
prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" |
|
924 |
proof - |
|
925 |
have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = |
|
926 |
prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))" |
|
927 |
by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool) |
|
928 |
also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))" |
|
929 |
using indep unfolding indep_var_def |
|
930 |
by (rule indep_varsD) (auto split: bool.split intro: sets) |
|
931 |
also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" |
|
932 |
unfolding UNIV_bool by simp |
|
933 |
finally show ?thesis . |
|
934 |
qed |
|
935 |
||
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
936 |
lemma (in prob_space) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
937 |
assumes "indep_var S X T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
938 |
shows indep_var_rv1: "random_variable S X" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
939 |
and indep_var_rv2: "random_variable T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
940 |
proof - |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
941 |
have "\<forall>i\<in>UNIV. random_variable (bool_case S T i) (bool_case X Y i)" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
942 |
using assms unfolding indep_var_def indep_vars_def by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
943 |
then show "random_variable S X" "random_variable T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
944 |
unfolding UNIV_bool by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
945 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
946 |
|
47694 | 947 |
lemma (in prob_space) indep_var_distribution_eq: |
948 |
"indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and> |
|
949 |
distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^isub>M ?T = ?J") |
|
950 |
proof safe |
|
951 |
assume "indep_var S X T Y" |
|
952 |
then show rvs: "random_variable S X" "random_variable T Y" |
|
953 |
by (blast dest: indep_var_rv1 indep_var_rv2)+ |
|
954 |
then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
955 |
by (rule measurable_Pair) |
|
956 |
||
957 |
interpret X: prob_space ?S by (rule prob_space_distr) fact |
|
958 |
interpret Y: prob_space ?T by (rule prob_space_distr) fact |
|
959 |
interpret XY: pair_prob_space ?S ?T .. |
|
960 |
show "?S \<Otimes>\<^isub>M ?T = ?J" |
|
961 |
proof (rule pair_measure_eqI) |
|
962 |
show "sigma_finite_measure ?S" .. |
|
963 |
show "sigma_finite_measure ?T" .. |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
964 |
|
47694 | 965 |
fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T" |
966 |
have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)" |
|
967 |
using A B by (intro emeasure_distr[OF XY]) auto |
|
968 |
also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)" |
|
969 |
using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure) |
|
970 |
also have "\<dots> = emeasure ?S A * emeasure ?T B" |
|
971 |
using rvs A B by (simp add: emeasure_distr) |
|
972 |
finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp |
|
973 |
qed simp |
|
974 |
next |
|
975 |
assume rvs: "random_variable S X" "random_variable T Y" |
|
976 |
then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
977 |
by (rule measurable_Pair) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
978 |
|
47694 | 979 |
let ?S = "distr M S X" and ?T = "distr M T Y" |
980 |
interpret X: prob_space ?S by (rule prob_space_distr) fact |
|
981 |
interpret Y: prob_space ?T by (rule prob_space_distr) fact |
|
982 |
interpret XY: pair_prob_space ?S ?T .. |
|
983 |
||
984 |
assume "?S \<Otimes>\<^isub>M ?T = ?J" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
985 |
|
47694 | 986 |
{ fix S and X |
987 |
have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}" |
|
988 |
proof (safe intro!: Int_stableI) |
|
989 |
fix A B assume "A \<in> sets S" "B \<in> sets S" |
|
990 |
then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S" |
|
991 |
by (intro exI[of _ "A \<inter> B"]) auto |
|
992 |
qed } |
|
993 |
note Int_stable = this |
|
994 |
||
995 |
show "indep_var S X T Y" unfolding indep_var_eq |
|
996 |
proof (intro conjI indep_set_sigma_sets Int_stable rvs) |
|
997 |
show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}" |
|
998 |
proof (safe intro!: indep_setI) |
|
999 |
{ fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M" |
|
1000 |
using `X \<in> measurable M S` by (auto intro: measurable_sets) } |
|
1001 |
{ fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M" |
|
1002 |
using `Y \<in> measurable M T` by (auto intro: measurable_sets) } |
|
1003 |
next |
|
1004 |
fix A B assume ab: "A \<in> sets S" "B \<in> sets T" |
|
1005 |
then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)" |
|
1006 |
using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"]) |
|
1007 |
also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)" |
|
1008 |
unfolding `?S \<Otimes>\<^isub>M ?T = ?J` .. |
|
1009 |
also have "\<dots> = emeasure ?S A * emeasure ?T B" |
|
49776 | 1010 |
using ab by (simp add: Y.emeasure_pair_measure_Times) |
47694 | 1011 |
finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) = |
1012 |
prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)" |
|
1013 |
using rvs ab by (simp add: emeasure_eq_measure emeasure_distr) |
|
1014 |
qed |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1015 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1016 |
qed |
42989 | 1017 |
|
49795 | 1018 |
lemma (in prob_space) distributed_joint_indep: |
1019 |
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" |
|
1020 |
assumes X: "distributed M S X Px" and Y: "distributed M T Y Py" |
|
1021 |
assumes indep: "indep_var S X T Y" |
|
1022 |
shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)" |
|
1023 |
using indep_var_distribution_eq[of S X T Y] indep |
|
1024 |
by (intro distributed_joint_indep'[OF S T X Y]) auto |
|
1025 |
||
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
1026 |
end |