author | hoelzl |
Thu, 26 May 2011 17:40:01 +0200 | |
changeset 42988 | d8f3fc934ff6 |
parent 42987 | 73e2d802ea41 |
child 42989 | 40adeda9a8d2 |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Independent_Family.thy |
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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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header {* Independent families of events, event sets, and random variables *} |
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theory Independent_Family |
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imports Probability_Measure |
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begin |
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|
42985 | 11 |
lemma INT_decseq_offset: |
12 |
assumes "decseq F" |
|
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shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)" |
|
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proof safe |
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fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)" |
|
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show "x \<in> F i" |
|
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proof cases |
|
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from x have "x \<in> F n" by auto |
|
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also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i" |
|
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unfolding decseq_def by simp |
|
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finally show ?thesis . |
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qed (insert x, simp) |
|
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qed auto |
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||
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definition (in prob_space) |
42983 | 26 |
"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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28 |
|
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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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31 |
|
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definition (in prob_space) |
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"indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and> |
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(\<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))))" |
35 |
||
36 |
definition (in prob_space) |
|
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"indep_set A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV" |
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38 |
|
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definition (in prob_space) |
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"indep_rv M' X I \<longleftrightarrow> |
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(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> |
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indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" |
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43 |
|
42987 | 44 |
lemma (in prob_space) indep_sets_cong[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" |
46 |
by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+ |
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||
42985 | 48 |
lemma (in prob_space) indep_sets_singleton_iff_indep_events: |
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"indep_sets (\<lambda>i. {F i}) I \<longleftrightarrow> indep_events F I" |
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unfolding indep_sets_def indep_events_def |
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by (simp, intro conj_cong ball_cong all_cong imp_cong) (auto simp: Pi_iff) |
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||
42981 | 53 |
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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68 |
|
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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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72 |
|
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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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76 |
shows "indep_sets G I" |
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proof - |
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78 |
have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)" |
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79 |
using mono by auto |
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80 |
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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81 |
using mono by (auto simp: Pi_iff) |
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82 |
ultimately show ?thesis |
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83 |
using indep by (auto simp: indep_sets_def) |
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84 |
qed |
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85 |
|
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86 |
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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89 |
shows "indep_sets F I" |
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90 |
using assms unfolding indep_sets_def by (auto simp: Pi_iff) |
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91 |
|
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lemma (in prob_space) indep_setsD: |
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93 |
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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94 |
shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
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95 |
using assms unfolding indep_sets_def by auto |
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96 |
|
42982 | 97 |
lemma (in prob_space) indep_setI: |
98 |
assumes ev: "A \<subseteq> events" "B \<subseteq> events" |
|
99 |
and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b" |
|
100 |
shows "indep_set A B" |
|
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unfolding indep_set_def |
|
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proof (rule indep_setsI) |
|
103 |
fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV" |
|
104 |
and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)" |
|
105 |
have "J \<in> Pow UNIV" by auto |
|
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with F `J \<noteq> {}` indep[of "F True" "F False"] |
|
107 |
show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))" |
|
108 |
unfolding UNIV_bool Pow_insert by (auto simp: ac_simps) |
|
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qed (auto split: bool.split simp: ev) |
|
110 |
||
111 |
lemma (in prob_space) indep_setD: |
|
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assumes indep: "indep_set A B" and ev: "a \<in> A" "b \<in> B" |
|
113 |
shows "prob (a \<inter> b) = prob a * prob b" |
|
114 |
using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case a b"] ev |
|
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by (simp add: ac_simps UNIV_bool) |
|
116 |
||
117 |
lemma (in prob_space) |
|
118 |
assumes indep: "indep_set A B" |
|
42983 | 119 |
shows indep_setD_ev1: "A \<subseteq> events" |
120 |
and indep_setD_ev2: "B \<subseteq> events" |
|
42982 | 121 |
using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto |
122 |
||
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123 |
lemma (in prob_space) indep_sets_dynkin: |
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124 |
assumes indep: "indep_sets F I" |
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125 |
shows "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) I" |
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126 |
(is "indep_sets ?F I") |
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127 |
proof (subst indep_sets_finite_index_sets, intro allI impI ballI) |
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128 |
fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}" |
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129 |
with indep have "indep_sets F J" |
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130 |
by (subst (asm) indep_sets_finite_index_sets) auto |
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131 |
{ fix J K assume "indep_sets F K" |
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132 |
let "?G S i" = "if i \<in> S then ?F i else F i" |
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133 |
assume "finite J" "J \<subseteq> K" |
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134 |
then have "indep_sets (?G J) K" |
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135 |
proof induct |
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136 |
case (insert j J) |
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137 |
moreover def G \<equiv> "?G J" |
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138 |
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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hoelzl
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139 |
by (auto simp: indep_sets_def) |
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140 |
let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }" |
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hoelzl
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141 |
{ fix X assume X: "X \<in> events" |
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142 |
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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143 |
\<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))" |
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144 |
have "indep_sets (G(j := {X})) K" |
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145 |
proof (rule indep_setsI) |
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|
146 |
fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events" |
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hoelzl
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|
147 |
using G X by auto |
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hoelzl
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|
148 |
next |
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149 |
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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150 |
show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" |
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|
151 |
proof cases |
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152 |
assume "j \<in> J" |
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|
153 |
with J have "A j = X" by auto |
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hoelzl
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|
154 |
show ?thesis |
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hoelzl
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|
155 |
proof cases |
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hoelzl
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|
156 |
assume "J = {j}" then show ?thesis by simp |
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hoelzl
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|
157 |
next |
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|
158 |
assume "J \<noteq> {j}" |
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Add formalization of probabilistic independence for families of sets
hoelzl
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|
159 |
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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160 |
using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm) |
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161 |
also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))" |
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162 |
proof (rule indep) |
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hoelzl
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|
163 |
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
|
164 |
using J `J \<noteq> {j}` `j \<in> J` by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
165 |
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
|
166 |
using J by auto |
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 |
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
|
169 |
using `A j = X` by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
170 |
also have "\<dots> = (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
171 |
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
|
172 |
using `j \<in> J` by (simp add: insert_absorb) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
173 |
finally show ?thesis . |
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 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
176 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
177 |
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
|
178 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
179 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
180 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
181 |
qed } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
182 |
note indep_sets_insert = this |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
183 |
have "dynkin_system \<lparr> space = space M, sets = ?D \<rparr>" |
42987 | 184 |
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
|
185 |
show "indep_sets (G(j := {{}})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
186 |
by (rule indep_sets_insert) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
187 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
188 |
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
|
189 |
show "indep_sets (G(j := {space M - X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
190 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
191 |
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
|
192 |
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
|
193 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
194 |
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) - (\<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 |
using A_sets sets_into_space X `J \<noteq> {}` |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
197 |
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
|
198 |
also have "\<dots> = 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
|
199 |
using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
200 |
by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
201 |
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
|
202 |
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
|
203 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
204 |
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
|
205 |
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
|
206 |
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
|
207 |
using prob_space by simp } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
208 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
209 |
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
|
210 |
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
|
211 |
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
|
212 |
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
|
213 |
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
|
214 |
by (simp add: field_simps) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
215 |
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
|
216 |
using X A by (simp add: finite_measure_compl) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
217 |
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
|
218 |
qed (insert X, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
219 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
220 |
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
|
221 |
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
|
222 |
show "indep_sets (G(j := {\<Union>k. F k})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
223 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
224 |
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
|
225 |
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
|
226 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
227 |
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
|
228 |
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
|
229 |
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
|
230 |
proof (rule finite_measure_UNION) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
231 |
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
|
232 |
using disj by (rule disjoint_family_on_bisimulation) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
233 |
show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
234 |
using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
235 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
236 |
moreover { fix k |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
237 |
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
|
238 |
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
|
239 |
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
|
240 |
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
|
241 |
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
|
242 |
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
|
243 |
by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
244 |
moreover |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
245 |
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
|
246 |
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
|
247 |
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
|
248 |
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
|
249 |
ultimately |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
250 |
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
|
251 |
by (auto dest!: sums_unique) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
252 |
qed (insert F, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
253 |
qed (insert sets_into_space, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
254 |
then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq> |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
255 |
sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>" |
42987 | 256 |
proof (rule dynkin_system.dynkin_subset, simp_all cong del: indep_sets_cong, safe) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
257 |
fix X assume "X \<in> G j" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
258 |
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
|
259 |
from `indep_sets G K` |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
260 |
show "indep_sets (G(j := {X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
261 |
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
|
262 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
263 |
have "indep_sets (G(j:=?D)) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
264 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
265 |
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
|
266 |
using G(2) by 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 |
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
|
269 |
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
|
270 |
proof cases |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
271 |
assume "j \<in> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
272 |
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
|
273 |
from J A show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
274 |
by (intro indep_setsD[OF indep]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
275 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
276 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
277 |
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
|
278 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
279 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
280 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
281 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
282 |
then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
283 |
by (rule indep_sets_mono_sets) (insert mono, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
284 |
then show ?case |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
285 |
by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
286 |
qed (insert `indep_sets F K`, simp) } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
287 |
from this[OF `indep_sets F J` `finite J` subset_refl] |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
288 |
show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
289 |
by (rule indep_sets_mono_sets) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
290 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
291 |
|
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
292 |
lemma (in prob_space) indep_sets_sigma: |
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hoelzl
parents:
diff
changeset
|
293 |
assumes indep: "indep_sets F I" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
294 |
assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
295 |
shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) I" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
296 |
proof - |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
297 |
from indep_sets_dynkin[OF indep] |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
298 |
show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
299 |
proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
300 |
fix i assume "i \<in> I" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
301 |
with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
302 |
with sets_into_space show "F i \<subseteq> Pow (space M)" by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
303 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
304 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
305 |
|
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
306 |
lemma (in prob_space) indep_sets_sigma_sets: |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
307 |
assumes "indep_sets F I" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
308 |
assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
309 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
310 |
using indep_sets_sigma[OF assms] by (simp add: sets_sigma) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
311 |
|
42987 | 312 |
lemma (in prob_space) indep_sets_sigma_sets_iff: |
313 |
assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>" |
|
314 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I" |
|
315 |
proof |
|
316 |
assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
|
317 |
by (rule indep_sets_sigma_sets) fact |
|
318 |
next |
|
319 |
assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I" |
|
320 |
by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic) |
|
321 |
qed |
|
322 |
||
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
323 |
lemma (in prob_space) indep_sets2_eq: |
42981 | 324 |
"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)" |
325 |
unfolding indep_set_def |
|
42861
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
326 |
proof (intro iffI ballI conjI) |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
327 |
assume indep: "indep_sets (bool_case A B) UNIV" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
328 |
{ fix a b assume "a \<in> A" "b \<in> B" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
329 |
with indep_setsD[OF indep, of UNIV "bool_case a b"] |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
330 |
show "prob (a \<inter> b) = prob a * prob b" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
331 |
unfolding UNIV_bool by (simp add: ac_simps) } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
332 |
from indep show "A \<subseteq> events" "B \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
333 |
unfolding indep_sets_def UNIV_bool by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
334 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
335 |
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)" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
336 |
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
|
337 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
338 |
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
|
339 |
using * by (auto split: bool.split) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
340 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
341 |
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)" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
342 |
then have "J = {True} \<or> J = {False} \<or> J = {True,False}" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
343 |
by (auto simp: UNIV_bool) |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
344 |
then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
345 |
using X * by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
346 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
347 |
qed |
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Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
348 |
|
42981 | 349 |
lemma (in prob_space) indep_set_sigma_sets: |
350 |
assumes "indep_set A B" |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
351 |
assumes A: "Int_stable \<lparr> space = space M, sets = A \<rparr>" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
352 |
assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>" |
42981 | 353 |
shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)" |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
354 |
proof - |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
355 |
have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
356 |
proof (rule indep_sets_sigma_sets) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
357 |
show "indep_sets (bool_case A B) UNIV" |
42981 | 358 |
by (rule `indep_set A B`[unfolded indep_set_def]) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
359 |
fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
360 |
using A B by (cases i) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
361 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
362 |
then show ?thesis |
42981 | 363 |
unfolding indep_set_def |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
364 |
by (rule indep_sets_mono_sets) (auto split: bool.split) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
365 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
366 |
|
42981 | 367 |
lemma (in prob_space) indep_sets_collect_sigma: |
368 |
fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set" |
|
369 |
assumes indep: "indep_sets E (\<Union>j\<in>J. I j)" |
|
370 |
assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable \<lparr>space = space M, sets = E i\<rparr>" |
|
371 |
assumes disjoint: "disjoint_family_on I J" |
|
372 |
shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J" |
|
373 |
proof - |
|
374 |
let "?E 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) }" |
|
375 |
||
42983 | 376 |
from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events" |
42981 | 377 |
unfolding indep_sets_def by auto |
378 |
{ fix j |
|
379 |
let ?S = "sigma \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>" |
|
380 |
assume "j \<in> J" |
|
381 |
from E[OF this] interpret S: sigma_algebra ?S |
|
382 |
using sets_into_space by (intro sigma_algebra_sigma) auto |
|
383 |
||
384 |
have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)" |
|
385 |
proof (rule sigma_sets_eqI) |
|
386 |
fix A assume "A \<in> (\<Union>i\<in>I j. E i)" |
|
387 |
then guess i .. |
|
388 |
then show "A \<in> sigma_sets (space M) (?E j)" |
|
389 |
by (auto intro!: sigma_sets.intros exI[of _ "{i}"] exI[of _ "\<lambda>i. A"]) |
|
390 |
next |
|
391 |
fix A assume "A \<in> ?E j" |
|
392 |
then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k" |
|
393 |
and A: "A = (\<Inter>k\<in>K. E' k)" |
|
394 |
by auto |
|
395 |
then have "A \<in> sets ?S" unfolding A |
|
396 |
by (safe intro!: S.finite_INT) |
|
397 |
(auto simp: sets_sigma intro!: sigma_sets.Basic) |
|
398 |
then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)" |
|
399 |
by (simp add: sets_sigma) |
|
400 |
qed } |
|
401 |
moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J" |
|
402 |
proof (rule indep_sets_sigma_sets) |
|
403 |
show "indep_sets ?E J" |
|
404 |
proof (intro indep_setsI) |
|
405 |
fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: finite_INT) |
|
406 |
next |
|
407 |
fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K" |
|
408 |
and "\<forall>j\<in>K. A j \<in> ?E j" |
|
409 |
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)" |
|
410 |
by simp |
|
411 |
from bchoice[OF this] guess E' .. |
|
412 |
from bchoice[OF this] obtain L |
|
413 |
where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)" |
|
414 |
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" |
|
415 |
and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l" |
|
416 |
by auto |
|
417 |
||
418 |
{ fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k" |
|
419 |
have "k = j" |
|
420 |
proof (rule ccontr) |
|
421 |
assume "k \<noteq> j" |
|
422 |
with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}" |
|
423 |
unfolding disjoint_family_on_def by auto |
|
424 |
with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`] |
|
425 |
show False using `l \<in> L k` `l \<in> L j` by auto |
|
426 |
qed } |
|
427 |
note L_inj = this |
|
428 |
||
429 |
def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)" |
|
430 |
{ fix x j l assume *: "j \<in> K" "l \<in> L j" |
|
431 |
have "k l = j" unfolding k_def |
|
432 |
proof (rule some_equality) |
|
433 |
fix k assume "k \<in> K \<and> l \<in> L k" |
|
434 |
with * L_inj show "k = j" by auto |
|
435 |
qed (insert *, simp) } |
|
436 |
note k_simp[simp] = this |
|
437 |
let "?E' l" = "E' (k l) l" |
|
438 |
have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)" |
|
439 |
by (auto simp: A intro!: arg_cong[where f=prob]) |
|
440 |
also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))" |
|
441 |
using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono) |
|
442 |
also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))" |
|
443 |
using K L L_inj by (subst setprod_UN_disjoint) auto |
|
444 |
also have "\<dots> = (\<Prod>j\<in>K. prob (A j))" |
|
445 |
using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast |
|
446 |
finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" . |
|
447 |
qed |
|
448 |
next |
|
449 |
fix j assume "j \<in> J" |
|
450 |
show "Int_stable \<lparr> space = space M, sets = ?E j \<rparr>" |
|
451 |
proof (rule Int_stableI) |
|
452 |
fix a assume "a \<in> ?E j" then obtain Ka Ea |
|
453 |
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 |
|
454 |
fix b assume "b \<in> ?E j" then obtain Kb Eb |
|
455 |
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 |
|
456 |
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 {})" |
|
457 |
have "a \<inter> b = INTER (Ka \<union> Kb) ?A" |
|
458 |
by (simp add: a b set_eq_iff) auto |
|
459 |
with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j" |
|
460 |
by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto |
|
461 |
qed |
|
462 |
qed |
|
463 |
ultimately show ?thesis |
|
464 |
by (simp cong: indep_sets_cong) |
|
465 |
qed |
|
466 |
||
42982 | 467 |
definition (in prob_space) terminal_events where |
468 |
"terminal_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
469 |
||
470 |
lemma (in prob_space) terminal_events_sets: |
|
42983 | 471 |
assumes A: "\<And>i. A i \<subseteq> events" |
42982 | 472 |
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>" |
473 |
assumes X: "X \<in> terminal_events A" |
|
42983 | 474 |
shows "X \<in> events" |
42982 | 475 |
proof - |
476 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
477 |
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact |
|
478 |
from X have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: terminal_events_def) |
|
479 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
|
42983 | 480 |
then show "X \<in> events" |
42982 | 481 |
by induct (insert A, auto) |
482 |
qed |
|
483 |
||
484 |
lemma (in prob_space) sigma_algebra_terminal_events: |
|
485 |
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>" |
|
486 |
shows "sigma_algebra \<lparr> space = space M, sets = terminal_events A \<rparr>" |
|
487 |
unfolding terminal_events_def |
|
488 |
proof (simp add: sigma_algebra_iff2, safe) |
|
489 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
490 |
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact |
|
491 |
{ fix X x assume "X \<in> ?A" "x \<in> X" |
|
492 |
then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto |
|
493 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
|
494 |
then have "X \<subseteq> space M" |
|
495 |
by induct (insert A.sets_into_space, auto) |
|
496 |
with `x \<in> X` show "x \<in> space M" by auto } |
|
497 |
{ fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A" |
|
498 |
then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)" |
|
499 |
by (intro sigma_sets.Union) auto } |
|
500 |
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty) |
|
501 |
||
502 |
lemma (in prob_space) kolmogorov_0_1_law: |
|
503 |
fixes A :: "nat \<Rightarrow> 'a set set" |
|
42983 | 504 |
assumes A: "\<And>i. A i \<subseteq> events" |
42982 | 505 |
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>" |
506 |
assumes indep: "indep_sets A UNIV" |
|
507 |
and X: "X \<in> terminal_events A" |
|
508 |
shows "prob X = 0 \<or> prob X = 1" |
|
509 |
proof - |
|
42983 | 510 |
let ?D = "\<lparr> space = space M, sets = {D \<in> events. prob (X \<inter> D) = prob X * prob D} \<rparr>" |
42982 | 511 |
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact |
512 |
interpret T: sigma_algebra "\<lparr> space = space M, sets = terminal_events A \<rparr>" |
|
513 |
by (rule sigma_algebra_terminal_events) fact |
|
514 |
have "X \<subseteq> space M" using T.space_closed X by auto |
|
515 |
||
42983 | 516 |
have X_in: "X \<in> events" |
42982 | 517 |
by (rule terminal_events_sets) fact+ |
518 |
||
519 |
interpret D: dynkin_system ?D |
|
520 |
proof (rule dynkin_systemI) |
|
521 |
fix D assume "D \<in> sets ?D" then show "D \<subseteq> space ?D" |
|
522 |
using sets_into_space by auto |
|
523 |
next |
|
524 |
show "space ?D \<in> sets ?D" |
|
525 |
using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2) |
|
526 |
next |
|
527 |
fix A assume A: "A \<in> sets ?D" |
|
528 |
have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))" |
|
529 |
using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob]) |
|
530 |
also have "\<dots> = prob X - prob (X \<inter> A)" |
|
531 |
using X_in A by (intro finite_measure_Diff) auto |
|
532 |
also have "\<dots> = prob X * prob (space M) - prob X * prob A" |
|
533 |
using A prob_space by auto |
|
534 |
also have "\<dots> = prob X * prob (space M - A)" |
|
535 |
using X_in A sets_into_space |
|
536 |
by (subst finite_measure_Diff) (auto simp: field_simps) |
|
537 |
finally show "space ?D - A \<in> sets ?D" |
|
538 |
using A `X \<subseteq> space M` by auto |
|
539 |
next |
|
540 |
fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> sets ?D" |
|
541 |
then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)" |
|
542 |
by auto |
|
543 |
have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)" |
|
544 |
proof (rule finite_measure_UNION) |
|
545 |
show "range (\<lambda>i. X \<inter> F i) \<subseteq> events" |
|
546 |
using F X_in by auto |
|
547 |
show "disjoint_family (\<lambda>i. X \<inter> F i)" |
|
548 |
using dis by (rule disjoint_family_on_bisimulation) auto |
|
549 |
qed |
|
550 |
with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))" |
|
551 |
by simp |
|
552 |
moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))" |
|
553 |
by (intro mult_right.sums finite_measure_UNION F dis) |
|
554 |
ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)" |
|
555 |
by (auto dest!: sums_unique) |
|
556 |
with F show "(\<Union>i. F i) \<in> sets ?D" |
|
557 |
by auto |
|
558 |
qed |
|
559 |
||
560 |
{ fix n |
|
561 |
have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV" |
|
562 |
proof (rule indep_sets_collect_sigma) |
|
563 |
have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _") |
|
564 |
by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq) |
|
565 |
with indep show "indep_sets A ?U" by simp |
|
566 |
show "disjoint_family (bool_case {..n} {Suc n..})" |
|
567 |
unfolding disjoint_family_on_def by (auto split: bool.split) |
|
568 |
fix m |
|
569 |
show "Int_stable \<lparr>space = space M, sets = A m\<rparr>" |
|
570 |
unfolding Int_stable_def using A.Int by auto |
|
571 |
qed |
|
572 |
also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) = |
|
573 |
bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))" |
|
574 |
by (auto intro!: ext split: bool.split) |
|
575 |
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))" |
|
576 |
unfolding indep_set_def by simp |
|
577 |
||
578 |
have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> sets ?D" |
|
579 |
proof (simp add: subset_eq, rule) |
|
580 |
fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
581 |
have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)" |
|
582 |
using X unfolding terminal_events_def by simp |
|
583 |
from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D |
|
584 |
show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D" |
|
585 |
by (auto simp add: ac_simps) |
|
586 |
qed } |
|
587 |
then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> sets ?D" (is "?A \<subseteq> _") |
|
588 |
by auto |
|
589 |
||
590 |
have "sigma \<lparr> space = space M, sets = ?A \<rparr> = |
|
591 |
dynkin \<lparr> space = space M, sets = ?A \<rparr>" (is "sigma ?UA = dynkin ?UA") |
|
592 |
proof (rule sigma_eq_dynkin) |
|
593 |
{ fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
594 |
then have "B \<subseteq> space M" |
|
595 |
by induct (insert A sets_into_space, auto) } |
|
596 |
then show "sets ?UA \<subseteq> Pow (space ?UA)" by auto |
|
597 |
show "Int_stable ?UA" |
|
598 |
proof (rule Int_stableI) |
|
599 |
fix a assume "a \<in> ?A" then guess n .. note a = this |
|
600 |
fix b assume "b \<in> ?A" then guess m .. note b = this |
|
601 |
interpret Amn: sigma_algebra "sigma \<lparr>space = space M, sets = (\<Union>i\<in>{..max m n}. A i)\<rparr>" |
|
602 |
using A sets_into_space by (intro sigma_algebra_sigma) auto |
|
603 |
have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
604 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
605 |
with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
606 |
moreover |
|
607 |
have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
608 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
609 |
with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
610 |
ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
611 |
using Amn.Int[of a b] by (simp add: sets_sigma) |
|
612 |
then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto |
|
613 |
qed |
|
614 |
qed |
|
615 |
moreover have "sets (dynkin ?UA) \<subseteq> sets ?D" |
|
616 |
proof (rule D.dynkin_subset) |
|
617 |
show "sets ?UA \<subseteq> sets ?D" using `?A \<subseteq> sets ?D` by auto |
|
618 |
qed simp |
|
619 |
ultimately have "sets (sigma ?UA) \<subseteq> sets ?D" by simp |
|
620 |
moreover |
|
621 |
have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A" |
|
622 |
by (intro sigma_sets_subseteq UN_mono) (auto intro: sigma_sets.Basic) |
|
623 |
then have "terminal_events A \<subseteq> sets (sigma ?UA)" |
|
624 |
unfolding sets_sigma terminal_events_def by auto |
|
625 |
moreover note `X \<in> terminal_events A` |
|
626 |
ultimately have "X \<in> sets ?D" by auto |
|
627 |
then show ?thesis by auto |
|
628 |
qed |
|
629 |
||
42985 | 630 |
lemma (in prob_space) borel_0_1_law: |
631 |
fixes F :: "nat \<Rightarrow> 'a set" |
|
632 |
assumes F: "range F \<subseteq> events" "indep_events F UNIV" |
|
633 |
shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1" |
|
634 |
proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"]) |
|
635 |
show "\<And>i. sigma_sets (space M) {F i} \<subseteq> events" |
|
636 |
using F(1) sets_into_space |
|
637 |
by (subst sigma_sets_singleton) auto |
|
638 |
{ fix i show "sigma_algebra \<lparr>space = space M, sets = sigma_sets (space M) {F i}\<rparr>" |
|
639 |
using sigma_algebra_sigma[of "\<lparr>space = space M, sets = {F i}\<rparr>"] F sets_into_space |
|
640 |
by (auto simp add: sigma_def) } |
|
641 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV" |
|
642 |
proof (rule indep_sets_sigma_sets) |
|
643 |
show "indep_sets (\<lambda>i. {F i}) UNIV" |
|
644 |
unfolding indep_sets_singleton_iff_indep_events by fact |
|
645 |
fix i show "Int_stable \<lparr>space = space M, sets = {F i}\<rparr>" |
|
646 |
unfolding Int_stable_def by simp |
|
647 |
qed |
|
648 |
let "?Q n" = "\<Union>i\<in>{n..}. F i" |
|
649 |
show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> terminal_events (\<lambda>i. sigma_sets (space M) {F i})" |
|
650 |
unfolding terminal_events_def |
|
651 |
proof |
|
652 |
fix j |
|
653 |
interpret S: sigma_algebra "sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>" |
|
654 |
using order_trans[OF F(1) space_closed] |
|
655 |
by (intro sigma_algebra_sigma) (simp add: sigma_sets_singleton subset_eq) |
|
656 |
have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)" |
|
657 |
by (intro decseq_SucI INT_decseq_offset UN_mono) auto |
|
658 |
also have "\<dots> \<in> sets (sigma \<lparr> space = space M, sets = (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})\<rparr>)" |
|
659 |
using order_trans[OF F(1) space_closed] |
|
660 |
by (safe intro!: S.countable_INT S.countable_UN) |
|
661 |
(auto simp: sets_sigma sigma_sets_singleton intro!: sigma_sets.Basic bexI) |
|
662 |
finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
|
663 |
by (simp add: sets_sigma) |
|
664 |
qed |
|
665 |
qed |
|
666 |
||
42987 | 667 |
lemma (in prob_space) indep_sets_finite: |
668 |
assumes I: "I \<noteq> {}" "finite I" |
|
669 |
and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" |
|
670 |
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)))" |
|
671 |
proof |
|
672 |
assume *: "indep_sets F I" |
|
673 |
from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
674 |
by (intro indep_setsD[OF *] ballI) auto |
|
675 |
next |
|
676 |
assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
677 |
show "indep_sets F I" |
|
678 |
proof (rule indep_setsI[OF F(1)]) |
|
679 |
fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" |
|
680 |
assume A: "\<forall>j\<in>J. A j \<in> F j" |
|
681 |
let "?A j" = "if j \<in> J then A j else space M" |
|
682 |
have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)" |
|
683 |
using subset_trans[OF F(1) space_closed] J A |
|
684 |
by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast |
|
685 |
also |
|
686 |
from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _") |
|
687 |
by (auto split: split_if_asm) |
|
688 |
with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))" |
|
689 |
by auto |
|
690 |
also have "\<dots> = (\<Prod>j\<in>J. prob (A j))" |
|
691 |
unfolding if_distrib setprod.If_cases[OF `finite I`] |
|
692 |
using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1) |
|
693 |
finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" .. |
|
694 |
qed |
|
695 |
qed |
|
696 |
||
697 |
lemma (in prob_space) indep_rv_finite: |
|
698 |
fixes I :: "'i set" |
|
699 |
assumes I: "I \<noteq> {}" "finite I" |
|
700 |
and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (sigma (M' i)) (X i)" |
|
701 |
and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (M' i)" |
|
702 |
and space: "\<And>i. i \<in> I \<Longrightarrow> space (M' i) \<in> sets (M' i)" |
|
703 |
shows "indep_rv (\<lambda>i. sigma (M' i)) X I \<longleftrightarrow> |
|
42988 | 704 |
(\<forall>A\<in>(\<Pi> i\<in>I. sets (M' 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 | 705 |
proof - |
706 |
from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)" |
|
707 |
unfolding measurable_def by simp |
|
708 |
||
709 |
{ fix i assume "i\<in>I" |
|
710 |
have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (sigma (M' i))} |
|
711 |
= sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
712 |
unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`] |
|
713 |
by (subst sigma_sets_sigma_sets_eq) auto } |
|
714 |
note this[simp] |
|
715 |
||
716 |
{ fix i assume "i\<in>I" |
|
717 |
have "Int_stable \<lparr>space = space M, sets = {X i -` A \<inter> space M |A. A \<in> sets (M' i)}\<rparr>" |
|
718 |
proof (rule Int_stableI) |
|
719 |
fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
720 |
then obtain A where "a = X i -` A \<inter> space M" "A \<in> sets (M' i)" by auto |
|
721 |
moreover |
|
722 |
fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
723 |
then obtain B where "b = X i -` B \<inter> space M" "B \<in> sets (M' i)" by auto |
|
724 |
moreover |
|
725 |
have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto |
|
726 |
moreover note Int_stable[OF `i \<in> I`] |
|
727 |
ultimately |
|
728 |
show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
729 |
by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD) |
|
730 |
qed } |
|
731 |
note indep_sets_sigma_sets_iff[OF this, simp] |
|
732 |
||
733 |
{ fix i assume "i \<in> I" |
|
734 |
{ fix A assume "A \<in> sets (M' i)" |
|
735 |
then have "A \<in> sets (sigma (M' i))" by (auto simp: sets_sigma intro: sigma_sets.Basic) |
|
736 |
moreover |
|
737 |
from rv[OF `i\<in>I`] have "X i \<in> measurable M (sigma (M' i))" by auto |
|
738 |
ultimately |
|
739 |
have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) } |
|
740 |
with X[OF `i\<in>I`] space[OF `i\<in>I`] |
|
741 |
have "{X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events" |
|
742 |
"space M \<in> {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
743 |
by (auto intro!: exI[of _ "space (M' i)"]) } |
|
744 |
note indep_sets_finite[OF I this, simp] |
|
745 |
||
746 |
have "(\<forall>A\<in>\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}. prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))) = |
|
747 |
(\<forall>A\<in>\<Pi> i\<in>I. sets (M' 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)))" |
|
748 |
(is "?L = ?R") |
|
749 |
proof safe |
|
750 |
fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. sets (M' i))" |
|
751 |
from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}` |
|
752 |
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))" |
|
753 |
by (auto simp add: Pi_iff) |
|
754 |
next |
|
755 |
fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> sets (M' i)})" |
|
756 |
from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> sets (M' i)" by auto |
|
757 |
from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M" |
|
758 |
"B \<in> (\<Pi> i\<in>I. sets (M' i))" by auto |
|
759 |
from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}` |
|
760 |
show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))" |
|
761 |
by simp |
|
762 |
qed |
|
763 |
then show ?thesis using `I \<noteq> {}` |
|
42988 | 764 |
by (simp add: rv indep_rv_def) |
765 |
qed |
|
766 |
||
767 |
lemma (in prob_space) indep_rv_compose: |
|
768 |
assumes "indep_rv M' X I" |
|
769 |
assumes rv: |
|
770 |
"\<And>i. i \<in> I \<Longrightarrow> sigma_algebra (N i)" |
|
771 |
"\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)" |
|
772 |
shows "indep_rv N (\<lambda>i. Y i \<circ> X i) I" |
|
773 |
unfolding indep_rv_def |
|
774 |
proof |
|
775 |
from rv `indep_rv M' X I` |
|
776 |
show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)" |
|
777 |
by (auto intro!: measurable_comp simp: indep_rv_def) |
|
778 |
||
779 |
have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
780 |
using `indep_rv M' X I` by (simp add: indep_rv_def) |
|
781 |
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" |
|
782 |
proof (rule indep_sets_mono_sets) |
|
783 |
fix i assume "i \<in> I" |
|
784 |
with `indep_rv M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)" |
|
785 |
unfolding indep_rv_def measurable_def by auto |
|
786 |
{ fix A assume "A \<in> sets (N i)" |
|
787 |
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)" |
|
788 |
by (intro exI[of _ "Y i -` A \<inter> space (M' i)"]) |
|
789 |
(auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) } |
|
790 |
then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq> |
|
791 |
sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
792 |
by (intro sigma_sets_subseteq) (auto simp: vimage_compose) |
|
793 |
qed |
|
794 |
qed |
|
795 |
||
796 |
lemma (in prob_space) indep_rvD: |
|
797 |
assumes X: "indep_rv M' X I" |
|
798 |
assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)" |
|
799 |
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))" |
|
800 |
proof (rule indep_setsD) |
|
801 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
802 |
using X by (auto simp: indep_rv_def) |
|
803 |
show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto |
|
804 |
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)}" |
|
805 |
using I by (auto intro: sigma_sets.Basic) |
|
806 |
qed |
|
807 |
||
808 |
lemma (in prob_space) indep_distribution_eq_measure: |
|
809 |
assumes I: "I \<noteq> {}" "finite I" |
|
810 |
assumes rv: "\<And>i. random_variable (M' i) (X i)" |
|
811 |
shows "indep_rv M' X I \<longleftrightarrow> |
|
812 |
(\<forall>A\<in>sets (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := extreal\<circ>distribution (X i) \<rparr>)). |
|
813 |
distribution (\<lambda>x. \<lambda>i\<in>I. X i x) A = |
|
814 |
finite_measure.\<mu>' (\<Pi>\<^isub>M i\<in>I. (M' i \<lparr> measure := extreal\<circ>distribution (X i) \<rparr>)) A)" |
|
815 |
(is "_ \<longleftrightarrow> (\<forall>X\<in>_. distribution ?D X = finite_measure.\<mu>' (Pi\<^isub>M I ?M) X)") |
|
816 |
proof - |
|
817 |
interpret M': prob_space "?M i" for i |
|
818 |
using rv by (rule distribution_prob_space) |
|
819 |
interpret P: finite_product_prob_space ?M I |
|
820 |
proof qed fact |
|
821 |
||
822 |
let ?D' = "(Pi\<^isub>M I ?M) \<lparr> measure := extreal \<circ> distribution ?D \<rparr>" |
|
823 |
have "random_variable P.P ?D" |
|
824 |
using `finite I` rv by (intro random_variable_restrict) auto |
|
825 |
then interpret D: prob_space ?D' |
|
826 |
by (rule distribution_prob_space) |
|
827 |
||
828 |
show ?thesis |
|
829 |
proof (intro iffI ballI) |
|
830 |
assume "indep_rv M' X I" |
|
831 |
fix A assume "A \<in> sets P.P" |
|
832 |
moreover |
|
833 |
have "D.prob A = P.prob A" |
|
834 |
proof (rule prob_space_unique_Int_stable) |
|
835 |
show "prob_space ?D'" by default |
|
836 |
show "prob_space (Pi\<^isub>M I ?M)" by default |
|
837 |
show "Int_stable P.G" using M'.Int |
|
838 |
by (intro Int_stable_product_algebra_generator) (simp add: Int_stable_def) |
|
839 |
show "space P.G \<in> sets P.G" |
|
840 |
using M'.top by (simp add: product_algebra_generator_def) |
|
841 |
show "space ?D' = space P.G" "sets ?D' = sets (sigma P.G)" |
|
842 |
by (simp_all add: product_algebra_def product_algebra_generator_def sets_sigma) |
|
843 |
show "space P.P = space P.G" "sets P.P = sets (sigma P.G)" |
|
844 |
by (simp_all add: product_algebra_def) |
|
845 |
show "A \<in> sets (sigma P.G)" |
|
846 |
using `A \<in> sets P.P` by (simp add: product_algebra_def) |
|
847 |
||
848 |
fix E assume E: "E \<in> sets P.G" |
|
849 |
then have "E \<in> sets P.P" |
|
850 |
by (simp add: sets_sigma sigma_sets.Basic product_algebra_def) |
|
851 |
then have "D.prob E = distribution ?D E" |
|
852 |
unfolding D.\<mu>'_def by simp |
|
853 |
also |
|
854 |
from E obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (M' i)" |
|
855 |
by (auto simp: product_algebra_generator_def) |
|
856 |
with `I \<noteq> {}` have "distribution ?D E = prob (\<Inter>i\<in>I. X i -` F i \<inter> space M)" |
|
857 |
using `I \<noteq> {}` by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def) |
|
858 |
also have "\<dots> = (\<Prod>i\<in>I. prob (X i -` F i \<inter> space M))" |
|
859 |
using `indep_rv M' X I` I F by (rule indep_rvD) |
|
860 |
also have "\<dots> = P.prob E" |
|
861 |
using F by (simp add: `E = Pi\<^isub>E I F` P.prob_times M'.\<mu>'_def distribution_def) |
|
862 |
finally show "D.prob E = P.prob E" . |
|
863 |
qed |
|
864 |
ultimately show "distribution ?D A = P.prob A" |
|
865 |
by (simp add: D.\<mu>'_def) |
|
866 |
next |
|
867 |
assume eq: "\<forall>A\<in>sets P.P. distribution ?D A = P.prob A" |
|
868 |
have [simp]: "\<And>i. sigma (M' i) = M' i" |
|
869 |
using rv by (intro sigma_algebra.sigma_eq) simp |
|
870 |
have "indep_rv (\<lambda>i. sigma (M' i)) X I" |
|
871 |
proof (subst indep_rv_finite[OF I]) |
|
872 |
fix i assume [simp]: "i \<in> I" |
|
873 |
show "random_variable (sigma (M' i)) (X i)" |
|
874 |
using rv[of i] by simp |
|
875 |
show "Int_stable (M' i)" "space (M' i) \<in> sets (M' i)" |
|
876 |
using M'.Int[of _ i] M'.top by (auto simp: Int_stable_def) |
|
877 |
next |
|
878 |
show "\<forall>A\<in>\<Pi> i\<in>I. sets (M' 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))" |
|
879 |
proof |
|
880 |
fix A assume A: "A \<in> (\<Pi> i\<in>I. sets (M' i))" |
|
881 |
then have A_in_P: "(Pi\<^isub>E I A) \<in> sets P.P" |
|
882 |
by (auto intro!: product_algebraI) |
|
883 |
have "prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = distribution ?D (Pi\<^isub>E I A)" |
|
884 |
using `I \<noteq> {}`by (auto intro!: arg_cong[where f=prob] simp: Pi_iff distribution_def) |
|
885 |
also have "\<dots> = P.prob (Pi\<^isub>E I A)" using A_in_P eq by simp |
|
886 |
also have "\<dots> = (\<Prod>i\<in>I. M'.prob i (A i))" |
|
887 |
using A by (intro P.prob_times) auto |
|
888 |
also have "\<dots> = (\<Prod>i\<in>I. prob (X i -` A i \<inter> space M))" |
|
889 |
using A by (auto intro!: setprod_cong simp: M'.\<mu>'_def Pi_iff distribution_def) |
|
890 |
finally show "prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M))" . |
|
891 |
qed |
|
892 |
qed |
|
893 |
then show "indep_rv M' X I" |
|
894 |
by simp |
|
895 |
qed |
|
42987 | 896 |
qed |
897 |
||
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
898 |
end |