author | hoelzl |
Wed, 10 Oct 2012 12:12:23 +0200 | |
changeset 49784 | 5e5b2da42a69 |
parent 49781 | 59b219ca3513 |
child 49794 | 3c7b5988e094 |
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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*) |
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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 Infinite_Product_Measure |
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begin |
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|
42985 | 11 |
lemma INT_decseq_offset: |
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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) |
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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))))" |
28 |
||
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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) |
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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 image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B" |
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by auto |
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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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definition (in prob_space) |
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"indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV" |
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definition (in prob_space) |
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"indep_vars 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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|
42989 | 55 |
definition (in prob_space) |
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"indep_var Ma A Mb B \<longleftrightarrow> indep_vars (bool_case Ma Mb) (bool_case A B) UNIV" |
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||
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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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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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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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|
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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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81 |
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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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|
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lemma (in prob_space) indep_sets_mono: |
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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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109 |
|
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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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using assms unfolding indep_sets_def by auto |
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114 |
|
42982 | 115 |
lemma (in prob_space) indep_setI: |
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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) |
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128 |
||
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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 |
|
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by (simp add: ac_simps UNIV_bool) |
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lemma (in prob_space) indep_var_eq: |
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"indep_var S X T Y \<longleftrightarrow> |
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(random_variable S X \<and> random_variable T Y) \<and> |
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138 |
indep_set |
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(sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S}) |
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(sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})" |
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141 |
unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool |
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142 |
by (intro arg_cong2[where f="op \<and>"] arg_cong2[where f=indep_sets] ext) |
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143 |
(auto split: bool.split) |
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144 |
|
42982 | 145 |
lemma (in prob_space) |
146 |
assumes indep: "indep_set A B" |
|
42983 | 147 |
shows indep_setD_ev1: "A \<subseteq> events" |
148 |
and indep_setD_ev2: "B \<subseteq> events" |
|
42982 | 149 |
using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto |
150 |
||
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lemma (in prob_space) indep_sets_dynkin: |
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152 |
assumes indep: "indep_sets F I" |
47694 | 153 |
shows "indep_sets (\<lambda>i. dynkin (space M) (F i)) I" |
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154 |
(is "indep_sets ?F I") |
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155 |
proof (subst indep_sets_finite_index_sets, intro allI impI ballI) |
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156 |
fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}" |
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157 |
with indep have "indep_sets F J" |
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by (subst (asm) indep_sets_finite_index_sets) auto |
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159 |
{ fix J K assume "indep_sets F K" |
46731 | 160 |
let ?G = "\<lambda>S i. if i \<in> S then ?F i else F i" |
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161 |
assume "finite J" "J \<subseteq> K" |
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162 |
then have "indep_sets (?G J) K" |
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163 |
proof induct |
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164 |
case (insert j J) |
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165 |
moreover def G \<equiv> "?G J" |
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|
166 |
ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
167 |
by (auto simp: indep_sets_def) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
168 |
let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
169 |
{ fix X assume X: "X \<in> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
170 |
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) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
171 |
\<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
172 |
have "indep_sets (G(j := {X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
173 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
174 |
fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
175 |
using G X by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
176 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
177 |
fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
178 |
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
|
179 |
proof cases |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
180 |
assume "j \<in> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
181 |
with J have "A j = X" by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
182 |
show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
183 |
proof cases |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
184 |
assume "J = {j}" then show ?thesis by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
185 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
186 |
assume "J \<noteq> {j}" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
187 |
have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
188 |
using `j \<in> J` `A j = X` 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
|
189 |
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
|
190 |
proof (rule indep) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
191 |
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
|
192 |
using J `J \<noteq> {j}` `j \<in> J` by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
193 |
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
|
194 |
using J by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
195 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
196 |
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
|
197 |
using `A j = X` by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
198 |
also have "\<dots> = (\<Prod>i\<in>J. prob (A i))" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
199 |
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
|
200 |
using `j \<in> J` by (simp add: insert_absorb) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
201 |
finally show ?thesis . |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
202 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
203 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
204 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
205 |
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
|
206 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
207 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
208 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
209 |
qed } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
210 |
note indep_sets_insert = this |
47694 | 211 |
have "dynkin_system (space M) ?D" |
42987 | 212 |
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
|
213 |
show "indep_sets (G(j := {{}})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
214 |
by (rule indep_sets_insert) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
215 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
216 |
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
|
217 |
show "indep_sets (G(j := {space M - X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
218 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
219 |
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
|
220 |
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
|
221 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
222 |
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
|
223 |
prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))" |
47694 | 224 |
using A_sets sets_into_space[of _ M] X `J \<noteq> {}` |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
225 |
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
|
226 |
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
|
227 |
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
|
228 |
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
|
229 |
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
|
230 |
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
|
231 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
232 |
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
|
233 |
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
|
234 |
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
|
235 |
using prob_space by simp } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
236 |
moreover { |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
237 |
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
|
238 |
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
|
239 |
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
|
240 |
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
|
241 |
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
|
242 |
by (simp add: field_simps) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
243 |
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
|
244 |
using X A by (simp add: finite_measure_compl) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
245 |
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
|
246 |
qed (insert X, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
247 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
248 |
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
|
249 |
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
|
250 |
show "indep_sets (G(j := {\<Union>k. F k})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
251 |
proof (rule indep_sets_insert) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
252 |
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
|
253 |
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
|
254 |
using G by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
255 |
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
|
256 |
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
|
257 |
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
|
258 |
proof (rule finite_measure_UNION) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
259 |
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
|
260 |
using disj by (rule disjoint_family_on_bisimulation) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
261 |
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
|
262 |
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
|
263 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
264 |
moreover { fix k |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
265 |
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
|
266 |
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
|
267 |
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
|
268 |
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
|
269 |
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
|
270 |
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
|
271 |
by simp |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
272 |
moreover |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
273 |
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
|
274 |
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
|
275 |
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
|
276 |
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
|
277 |
ultimately |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
278 |
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
|
279 |
by (auto dest!: sums_unique) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
280 |
qed (insert F, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
281 |
qed (insert sets_into_space, auto) |
47694 | 282 |
then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}" |
283 |
proof (rule dynkin_system.dynkin_subset, safe) |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
284 |
fix X assume "X \<in> G j" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
285 |
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
|
286 |
from `indep_sets G K` |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
287 |
show "indep_sets (G(j := {X})) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
288 |
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
|
289 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
290 |
have "indep_sets (G(j:=?D)) K" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
291 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
292 |
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
|
293 |
using G(2) by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
294 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
295 |
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
|
296 |
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
|
297 |
proof cases |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
298 |
assume "j \<in> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
299 |
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
|
300 |
from J A show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
301 |
by (intro indep_setsD[OF indep]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
302 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
303 |
assume "j \<notin> J" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
304 |
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
|
305 |
with J show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
306 |
by (intro indep_setsD[OF G(1)]) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
307 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
308 |
qed |
47694 | 309 |
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
|
310 |
by (rule indep_sets_mono_sets) (insert mono, auto) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
311 |
then show ?case |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
312 |
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
|
313 |
qed (insert `indep_sets F K`, simp) } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
314 |
from this[OF `indep_sets F J` `finite J` subset_refl] |
47694 | 315 |
show "indep_sets ?F J" |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
316 |
by (rule indep_sets_mono_sets) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
317 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
318 |
|
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
319 |
lemma (in prob_space) indep_sets_sigma: |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
320 |
assumes indep: "indep_sets F I" |
47694 | 321 |
assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" |
322 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
323 |
proof - |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
324 |
from indep_sets_dynkin[OF indep] |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
325 |
show ?thesis |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
326 |
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
|
327 |
fix i assume "i \<in> I" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
328 |
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
|
329 |
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
|
330 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
331 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
332 |
|
42987 | 333 |
lemma (in prob_space) indep_sets_sigma_sets_iff: |
47694 | 334 |
assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" |
42987 | 335 |
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I \<longleftrightarrow> indep_sets F I" |
336 |
proof |
|
337 |
assume "indep_sets F I" then show "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" |
|
47694 | 338 |
by (rule indep_sets_sigma) fact |
42987 | 339 |
next |
340 |
assume "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I" then show "indep_sets F I" |
|
341 |
by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic) |
|
342 |
qed |
|
343 |
||
49781 | 344 |
lemma (in prob_space) |
345 |
"indep_vars M' X I \<longleftrightarrow> |
|
346 |
(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> |
|
347 |
indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I" |
|
348 |
unfolding indep_vars_def |
|
349 |
apply (rule conj_cong[OF refl]) |
|
350 |
apply (rule indep_sets_sigma_sets_iff) |
|
351 |
apply (auto simp: Int_stable_def) |
|
352 |
apply (rule_tac x="A \<inter> Aa" in exI) |
|
353 |
apply auto |
|
354 |
done |
|
355 |
||
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
356 |
lemma (in prob_space) indep_sets2_eq: |
42981 | 357 |
"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)" |
358 |
unfolding indep_set_def |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
359 |
proof (intro iffI ballI conjI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
360 |
assume indep: "indep_sets (bool_case A B) UNIV" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
361 |
{ fix a b assume "a \<in> A" "b \<in> B" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
362 |
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
|
363 |
show "prob (a \<inter> b) = prob a * prob b" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
364 |
unfolding UNIV_bool by (simp add: ac_simps) } |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
365 |
from indep show "A \<subseteq> events" "B \<subseteq> events" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
366 |
unfolding indep_sets_def UNIV_bool by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
367 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
368 |
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
|
369 |
show "indep_sets (bool_case A B) UNIV" |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
370 |
proof (rule indep_setsI) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
371 |
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
|
372 |
using * by (auto split: bool.split) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
373 |
next |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
374 |
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
|
375 |
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
|
376 |
by (auto simp: UNIV_bool) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
377 |
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
|
378 |
using X * by auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
379 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
380 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
381 |
|
42981 | 382 |
lemma (in prob_space) indep_set_sigma_sets: |
383 |
assumes "indep_set A B" |
|
47694 | 384 |
assumes A: "Int_stable A" and B: "Int_stable B" |
42981 | 385 |
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
|
386 |
proof - |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
387 |
have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV" |
47694 | 388 |
proof (rule indep_sets_sigma) |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
389 |
show "indep_sets (bool_case A B) UNIV" |
42981 | 390 |
by (rule `indep_set A B`[unfolded indep_set_def]) |
47694 | 391 |
fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)" |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
392 |
using A B by (cases i) auto |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
393 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
394 |
then show ?thesis |
42981 | 395 |
unfolding indep_set_def |
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
396 |
by (rule indep_sets_mono_sets) (auto split: bool.split) |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
397 |
qed |
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
398 |
|
42981 | 399 |
lemma (in prob_space) indep_sets_collect_sigma: |
400 |
fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set" |
|
401 |
assumes indep: "indep_sets E (\<Union>j\<in>J. I j)" |
|
47694 | 402 |
assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable (E i)" |
42981 | 403 |
assumes disjoint: "disjoint_family_on I J" |
404 |
shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J" |
|
405 |
proof - |
|
46731 | 406 |
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 | 407 |
|
42983 | 408 |
from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> events" |
42981 | 409 |
unfolding indep_sets_def by auto |
410 |
{ fix j |
|
47694 | 411 |
let ?S = "sigma_sets (space M) (\<Union>i\<in>I j. E i)" |
42981 | 412 |
assume "j \<in> J" |
47694 | 413 |
from E[OF this] interpret S: sigma_algebra "space M" ?S |
414 |
using sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto |
|
42981 | 415 |
|
416 |
have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)" |
|
417 |
proof (rule sigma_sets_eqI) |
|
418 |
fix A assume "A \<in> (\<Union>i\<in>I j. E i)" |
|
419 |
then guess i .. |
|
420 |
then show "A \<in> sigma_sets (space M) (?E j)" |
|
47694 | 421 |
by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\<lambda>i. A"]) |
42981 | 422 |
next |
423 |
fix A assume "A \<in> ?E j" |
|
424 |
then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k" |
|
425 |
and A: "A = (\<Inter>k\<in>K. E' k)" |
|
426 |
by auto |
|
47694 | 427 |
then have "A \<in> ?S" unfolding A |
428 |
by (safe intro!: S.finite_INT) auto |
|
42981 | 429 |
then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)" |
47694 | 430 |
by simp |
42981 | 431 |
qed } |
432 |
moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J" |
|
47694 | 433 |
proof (rule indep_sets_sigma) |
42981 | 434 |
show "indep_sets ?E J" |
435 |
proof (intro indep_setsI) |
|
436 |
fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: finite_INT) |
|
437 |
next |
|
438 |
fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K" |
|
439 |
and "\<forall>j\<in>K. A j \<in> ?E j" |
|
440 |
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)" |
|
441 |
by simp |
|
442 |
from bchoice[OF this] guess E' .. |
|
443 |
from bchoice[OF this] obtain L |
|
444 |
where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)" |
|
445 |
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" |
|
446 |
and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l" |
|
447 |
by auto |
|
448 |
||
449 |
{ fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k" |
|
450 |
have "k = j" |
|
451 |
proof (rule ccontr) |
|
452 |
assume "k \<noteq> j" |
|
453 |
with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}" |
|
454 |
unfolding disjoint_family_on_def by auto |
|
455 |
with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`] |
|
456 |
show False using `l \<in> L k` `l \<in> L j` by auto |
|
457 |
qed } |
|
458 |
note L_inj = this |
|
459 |
||
460 |
def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)" |
|
461 |
{ fix x j l assume *: "j \<in> K" "l \<in> L j" |
|
462 |
have "k l = j" unfolding k_def |
|
463 |
proof (rule some_equality) |
|
464 |
fix k assume "k \<in> K \<and> l \<in> L k" |
|
465 |
with * L_inj show "k = j" by auto |
|
466 |
qed (insert *, simp) } |
|
467 |
note k_simp[simp] = this |
|
46731 | 468 |
let ?E' = "\<lambda>l. E' (k l) l" |
42981 | 469 |
have "prob (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)" |
470 |
by (auto simp: A intro!: arg_cong[where f=prob]) |
|
471 |
also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))" |
|
472 |
using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono) |
|
473 |
also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))" |
|
474 |
using K L L_inj by (subst setprod_UN_disjoint) auto |
|
475 |
also have "\<dots> = (\<Prod>j\<in>K. prob (A j))" |
|
476 |
using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast |
|
477 |
finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" . |
|
478 |
qed |
|
479 |
next |
|
480 |
fix j assume "j \<in> J" |
|
47694 | 481 |
show "Int_stable (?E j)" |
42981 | 482 |
proof (rule Int_stableI) |
483 |
fix a assume "a \<in> ?E j" then obtain Ka Ea |
|
484 |
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 |
|
485 |
fix b assume "b \<in> ?E j" then obtain Kb Eb |
|
486 |
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 |
|
487 |
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 {})" |
|
488 |
have "a \<inter> b = INTER (Ka \<union> Kb) ?A" |
|
489 |
by (simp add: a b set_eq_iff) auto |
|
490 |
with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j" |
|
491 |
by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto |
|
492 |
qed |
|
493 |
qed |
|
494 |
ultimately show ?thesis |
|
495 |
by (simp cong: indep_sets_cong) |
|
496 |
qed |
|
497 |
||
49772 | 498 |
definition (in prob_space) tail_events where |
499 |
"tail_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
42982 | 500 |
|
49772 | 501 |
lemma (in prob_space) tail_events_sets: |
502 |
assumes A: "\<And>i::nat. A i \<subseteq> events" |
|
503 |
shows "tail_events A \<subseteq> events" |
|
504 |
proof |
|
505 |
fix X assume X: "X \<in> tail_events A" |
|
42982 | 506 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
49772 | 507 |
from X have "\<And>n::nat. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: tail_events_def) |
42982 | 508 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
42983 | 509 |
then show "X \<in> events" |
42982 | 510 |
by induct (insert A, auto) |
511 |
qed |
|
512 |
||
49772 | 513 |
lemma (in prob_space) sigma_algebra_tail_events: |
47694 | 514 |
assumes "\<And>i::nat. sigma_algebra (space M) (A i)" |
49772 | 515 |
shows "sigma_algebra (space M) (tail_events A)" |
516 |
unfolding tail_events_def |
|
42982 | 517 |
proof (simp add: sigma_algebra_iff2, safe) |
518 |
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))" |
|
47694 | 519 |
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
|
520 |
{ fix X x assume "X \<in> ?A" "x \<in> X" |
42982 | 521 |
then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto |
522 |
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp |
|
523 |
then have "X \<subseteq> space M" |
|
524 |
by induct (insert A.sets_into_space, auto) |
|
525 |
with `x \<in> X` show "x \<in> space M" by auto } |
|
526 |
{ fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A" |
|
527 |
then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)" |
|
528 |
by (intro sigma_sets.Union) auto } |
|
529 |
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty) |
|
530 |
||
531 |
lemma (in prob_space) kolmogorov_0_1_law: |
|
532 |
fixes A :: "nat \<Rightarrow> 'a set set" |
|
47694 | 533 |
assumes "\<And>i::nat. sigma_algebra (space M) (A i)" |
42982 | 534 |
assumes indep: "indep_sets A UNIV" |
49772 | 535 |
and X: "X \<in> tail_events A" |
42982 | 536 |
shows "prob X = 0 \<or> prob X = 1" |
537 |
proof - |
|
49781 | 538 |
have A: "\<And>i. A i \<subseteq> events" |
539 |
using indep unfolding indep_sets_def by simp |
|
540 |
||
47694 | 541 |
let ?D = "{D \<in> events. prob (X \<inter> D) = prob X * prob D}" |
542 |
interpret A: sigma_algebra "space M" "A i" for i by fact |
|
49772 | 543 |
interpret T: sigma_algebra "space M" "tail_events A" |
544 |
by (rule sigma_algebra_tail_events) fact |
|
42982 | 545 |
have "X \<subseteq> space M" using T.space_closed X by auto |
546 |
||
42983 | 547 |
have X_in: "X \<in> events" |
49772 | 548 |
using tail_events_sets A X by auto |
42982 | 549 |
|
47694 | 550 |
interpret D: dynkin_system "space M" ?D |
42982 | 551 |
proof (rule dynkin_systemI) |
47694 | 552 |
fix D assume "D \<in> ?D" then show "D \<subseteq> space M" |
42982 | 553 |
using sets_into_space by auto |
554 |
next |
|
47694 | 555 |
show "space M \<in> ?D" |
42982 | 556 |
using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2) |
557 |
next |
|
47694 | 558 |
fix A assume A: "A \<in> ?D" |
42982 | 559 |
have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))" |
560 |
using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob]) |
|
561 |
also have "\<dots> = prob X - prob (X \<inter> A)" |
|
562 |
using X_in A by (intro finite_measure_Diff) auto |
|
563 |
also have "\<dots> = prob X * prob (space M) - prob X * prob A" |
|
564 |
using A prob_space by auto |
|
565 |
also have "\<dots> = prob X * prob (space M - A)" |
|
566 |
using X_in A sets_into_space |
|
567 |
by (subst finite_measure_Diff) (auto simp: field_simps) |
|
47694 | 568 |
finally show "space M - A \<in> ?D" |
42982 | 569 |
using A `X \<subseteq> space M` by auto |
570 |
next |
|
47694 | 571 |
fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D" |
42982 | 572 |
then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)" |
573 |
by auto |
|
574 |
have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)" |
|
575 |
proof (rule finite_measure_UNION) |
|
576 |
show "range (\<lambda>i. X \<inter> F i) \<subseteq> events" |
|
577 |
using F X_in by auto |
|
578 |
show "disjoint_family (\<lambda>i. X \<inter> F i)" |
|
579 |
using dis by (rule disjoint_family_on_bisimulation) auto |
|
580 |
qed |
|
581 |
with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))" |
|
582 |
by simp |
|
583 |
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
|
584 |
by (intro sums_mult finite_measure_UNION F dis) |
42982 | 585 |
ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)" |
586 |
by (auto dest!: sums_unique) |
|
47694 | 587 |
with F show "(\<Union>i. F i) \<in> ?D" |
42982 | 588 |
by auto |
589 |
qed |
|
590 |
||
591 |
{ fix n |
|
592 |
have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV" |
|
593 |
proof (rule indep_sets_collect_sigma) |
|
594 |
have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {Suc n..}) = UNIV" (is "?U = _") |
|
595 |
by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq) |
|
596 |
with indep show "indep_sets A ?U" by simp |
|
597 |
show "disjoint_family (bool_case {..n} {Suc n..})" |
|
598 |
unfolding disjoint_family_on_def by (auto split: bool.split) |
|
599 |
fix m |
|
47694 | 600 |
show "Int_stable (A m)" |
42982 | 601 |
unfolding Int_stable_def using A.Int by auto |
602 |
qed |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
603 |
also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) = |
42982 | 604 |
bool_case (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))" |
605 |
by (auto intro!: ext split: bool.split) |
|
606 |
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))" |
|
607 |
unfolding indep_set_def by simp |
|
608 |
||
47694 | 609 |
have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> ?D" |
42982 | 610 |
proof (simp add: subset_eq, rule) |
611 |
fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
612 |
have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)" |
|
49772 | 613 |
using X unfolding tail_events_def by simp |
42982 | 614 |
from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D |
615 |
show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D" |
|
616 |
by (auto simp add: ac_simps) |
|
617 |
qed } |
|
47694 | 618 |
then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _") |
42982 | 619 |
by auto |
620 |
||
49772 | 621 |
note `X \<in> tail_events A` |
47694 | 622 |
also { |
623 |
have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A" |
|
624 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
49772 | 625 |
then have "tail_events A \<subseteq> sigma_sets (space M) ?A" |
626 |
unfolding tail_events_def by auto } |
|
47694 | 627 |
also have "sigma_sets (space M) ?A = dynkin (space M) ?A" |
42982 | 628 |
proof (rule sigma_eq_dynkin) |
629 |
{ fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)" |
|
630 |
then have "B \<subseteq> space M" |
|
47694 | 631 |
by induct (insert A sets_into_space[of _ M], auto) } |
632 |
then show "?A \<subseteq> Pow (space M)" by auto |
|
633 |
show "Int_stable ?A" |
|
42982 | 634 |
proof (rule Int_stableI) |
635 |
fix a assume "a \<in> ?A" then guess n .. note a = this |
|
636 |
fix b assume "b \<in> ?A" then guess m .. note b = this |
|
47694 | 637 |
interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
638 |
using A sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto |
|
42982 | 639 |
have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
640 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
641 |
with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
642 |
moreover |
|
643 |
have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
644 |
by (intro sigma_sets_subseteq UN_mono) auto |
|
645 |
with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto |
|
646 |
ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" |
|
47694 | 647 |
using Amn.Int[of a b] by simp |
42982 | 648 |
then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto |
649 |
qed |
|
650 |
qed |
|
47694 | 651 |
also have "dynkin (space M) ?A \<subseteq> ?D" |
652 |
using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset) |
|
653 |
finally show ?thesis by auto |
|
42982 | 654 |
qed |
655 |
||
42985 | 656 |
lemma (in prob_space) borel_0_1_law: |
657 |
fixes F :: "nat \<Rightarrow> 'a set" |
|
49781 | 658 |
assumes F2: "indep_events F UNIV" |
42985 | 659 |
shows "prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 0 \<or> prob (\<Inter>n. \<Union>m\<in>{n..}. F m) = 1" |
660 |
proof (rule kolmogorov_0_1_law[of "\<lambda>i. sigma_sets (space M) { F i }"]) |
|
49781 | 661 |
have F1: "range F \<subseteq> events" |
662 |
using F2 by (simp add: indep_events_def subset_eq) |
|
47694 | 663 |
{ fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})" |
49781 | 664 |
using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets_into_space |
47694 | 665 |
by auto } |
42985 | 666 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {F i}) UNIV" |
47694 | 667 |
proof (rule indep_sets_sigma) |
42985 | 668 |
show "indep_sets (\<lambda>i. {F i}) UNIV" |
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49781
diff
changeset
|
669 |
unfolding indep_events_def_alt[symmetric] by fact |
47694 | 670 |
fix i show "Int_stable {F i}" |
42985 | 671 |
unfolding Int_stable_def by simp |
672 |
qed |
|
46731 | 673 |
let ?Q = "\<lambda>n. \<Union>i\<in>{n..}. F i" |
49772 | 674 |
show "(\<Inter>n. \<Union>m\<in>{n..}. F m) \<in> tail_events (\<lambda>i. sigma_sets (space M) {F i})" |
675 |
unfolding tail_events_def |
|
42985 | 676 |
proof |
677 |
fix j |
|
47694 | 678 |
interpret S: sigma_algebra "space M" "sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
49781 | 679 |
using order_trans[OF F1 space_closed] |
47694 | 680 |
by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq) |
42985 | 681 |
have "(\<Inter>n. ?Q n) = (\<Inter>n\<in>{j..}. ?Q n)" |
682 |
by (intro decseq_SucI INT_decseq_offset UN_mono) auto |
|
47694 | 683 |
also have "\<dots> \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
49781 | 684 |
using order_trans[OF F1 space_closed] |
42985 | 685 |
by (safe intro!: S.countable_INT S.countable_UN) |
47694 | 686 |
(auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI) |
42985 | 687 |
finally show "(\<Inter>n. ?Q n) \<in> sigma_sets (space M) (\<Union>i\<in>{j..}. sigma_sets (space M) {F i})" |
47694 | 688 |
by simp |
42985 | 689 |
qed |
690 |
qed |
|
691 |
||
42987 | 692 |
lemma (in prob_space) indep_sets_finite: |
693 |
assumes I: "I \<noteq> {}" "finite I" |
|
694 |
and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events" "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" |
|
695 |
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)))" |
|
696 |
proof |
|
697 |
assume *: "indep_sets F I" |
|
698 |
from I show "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
699 |
by (intro indep_setsD[OF *] ballI) auto |
|
700 |
next |
|
701 |
assume indep: "\<forall>A\<in>Pi I F. prob (\<Inter>j\<in>I. A j) = (\<Prod>j\<in>I. prob (A j))" |
|
702 |
show "indep_sets F I" |
|
703 |
proof (rule indep_setsI[OF F(1)]) |
|
704 |
fix A J assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" |
|
705 |
assume A: "\<forall>j\<in>J. A j \<in> F j" |
|
46731 | 706 |
let ?A = "\<lambda>j. if j \<in> J then A j else space M" |
42987 | 707 |
have "prob (\<Inter>j\<in>I. ?A j) = prob (\<Inter>j\<in>J. A j)" |
708 |
using subset_trans[OF F(1) space_closed] J A |
|
709 |
by (auto intro!: arg_cong[where f=prob] split: split_if_asm) blast |
|
710 |
also |
|
711 |
from A F have "(\<lambda>j. if j \<in> J then A j else space M) \<in> Pi I F" (is "?A \<in> _") |
|
712 |
by (auto split: split_if_asm) |
|
713 |
with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))" |
|
714 |
by auto |
|
715 |
also have "\<dots> = (\<Prod>j\<in>J. prob (A j))" |
|
716 |
unfolding if_distrib setprod.If_cases[OF `finite I`] |
|
717 |
using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod_1) |
|
718 |
finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" .. |
|
719 |
qed |
|
720 |
qed |
|
721 |
||
42989 | 722 |
lemma (in prob_space) indep_vars_finite: |
42987 | 723 |
fixes I :: "'i set" |
724 |
assumes I: "I \<noteq> {}" "finite I" |
|
47694 | 725 |
and M': "\<And>i. i \<in> I \<Longrightarrow> sets (M' i) = sigma_sets (space (M' i)) (E i)" |
726 |
and rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)" |
|
727 |
and Int_stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable (E i)" |
|
728 |
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))" |
|
729 |
shows "indep_vars M' X I \<longleftrightarrow> |
|
730 |
(\<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 | 731 |
proof - |
732 |
from rv have X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> space M \<rightarrow> space (M' i)" |
|
733 |
unfolding measurable_def by simp |
|
734 |
||
735 |
{ fix i assume "i\<in>I" |
|
47694 | 736 |
from closed[OF `i \<in> I`] |
737 |
have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} |
|
738 |
= sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}" |
|
739 |
unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`] |
|
42987 | 740 |
by (subst sigma_sets_sigma_sets_eq) auto } |
47694 | 741 |
note sigma_sets_X = this |
42987 | 742 |
|
743 |
{ fix i assume "i\<in>I" |
|
47694 | 744 |
have "Int_stable {X i -` A \<inter> space M |A. A \<in> E i}" |
42987 | 745 |
proof (rule Int_stableI) |
47694 | 746 |
fix a assume "a \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
747 |
then obtain A where "a = X i -` A \<inter> space M" "A \<in> E i" by auto |
|
42987 | 748 |
moreover |
47694 | 749 |
fix b assume "b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
750 |
then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto |
|
42987 | 751 |
moreover |
752 |
have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto |
|
753 |
moreover note Int_stable[OF `i \<in> I`] |
|
754 |
ultimately |
|
47694 | 755 |
show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
42987 | 756 |
by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD) |
757 |
qed } |
|
47694 | 758 |
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
|
759 |
|
42987 | 760 |
{ fix i assume "i \<in> I" |
47694 | 761 |
{ fix A assume "A \<in> E i" |
762 |
with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto |
|
42987 | 763 |
moreover |
47694 | 764 |
from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto |
42987 | 765 |
ultimately |
766 |
have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) } |
|
767 |
with X[OF `i\<in>I`] space[OF `i\<in>I`] |
|
47694 | 768 |
have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events" |
769 |
"space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}" |
|
42987 | 770 |
by (auto intro!: exI[of _ "space (M' i)"]) } |
47694 | 771 |
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
|
772 |
|
47694 | 773 |
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))) = |
774 |
(\<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 | 775 |
(is "?L = ?R") |
776 |
proof safe |
|
47694 | 777 |
fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)" |
42987 | 778 |
from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}` |
779 |
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))" |
|
780 |
by (auto simp add: Pi_iff) |
|
781 |
next |
|
47694 | 782 |
fix A assume ?R and A: "A \<in> (\<Pi> i\<in>I. {X i -` A \<inter> space M |A. A \<in> E i})" |
783 |
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 | 784 |
from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M" |
47694 | 785 |
"B \<in> (\<Pi> i\<in>I. E i)" by auto |
42987 | 786 |
from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}` |
787 |
show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))" |
|
788 |
by simp |
|
789 |
qed |
|
790 |
then show ?thesis using `I \<noteq> {}` |
|
47694 | 791 |
by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong) |
42988 | 792 |
qed |
793 |
||
42989 | 794 |
lemma (in prob_space) indep_vars_compose: |
795 |
assumes "indep_vars M' X I" |
|
47694 | 796 |
assumes rv: "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> measurable (M' i) (N i)" |
42989 | 797 |
shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I" |
798 |
unfolding indep_vars_def |
|
42988 | 799 |
proof |
42989 | 800 |
from rv `indep_vars M' X I` |
42988 | 801 |
show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)" |
47694 | 802 |
by (auto simp: indep_vars_def) |
42988 | 803 |
|
804 |
have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
42989 | 805 |
using `indep_vars M' X I` by (simp add: indep_vars_def) |
42988 | 806 |
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" |
807 |
proof (rule indep_sets_mono_sets) |
|
808 |
fix i assume "i \<in> I" |
|
42989 | 809 |
with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)" |
810 |
unfolding indep_vars_def measurable_def by auto |
|
42988 | 811 |
{ fix A assume "A \<in> sets (N i)" |
812 |
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)" |
|
813 |
by (intro exI[of _ "Y i -` A \<inter> space (M' i)"]) |
|
814 |
(auto simp: vimage_compose intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) } |
|
815 |
then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq> |
|
816 |
sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}" |
|
817 |
by (intro sigma_sets_subseteq) (auto simp: vimage_compose) |
|
818 |
qed |
|
819 |
qed |
|
820 |
||
47694 | 821 |
lemma (in prob_space) indep_varsD_finite: |
42989 | 822 |
assumes X: "indep_vars M' X I" |
42988 | 823 |
assumes I: "I \<noteq> {}" "finite I" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M' i)" |
824 |
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))" |
|
825 |
proof (rule indep_setsD) |
|
826 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
42989 | 827 |
using X by (auto simp: indep_vars_def) |
42988 | 828 |
show "I \<subseteq> I" "I \<noteq> {}" "finite I" using I by auto |
829 |
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 | 830 |
using I by auto |
42988 | 831 |
qed |
832 |
||
47694 | 833 |
lemma (in prob_space) indep_varsD: |
834 |
assumes X: "indep_vars M' X I" |
|
835 |
assumes I: "J \<noteq> {}" "finite J" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)" |
|
836 |
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))" |
|
837 |
proof (rule indep_setsD) |
|
838 |
show "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I" |
|
839 |
using X by (auto simp: indep_vars_def) |
|
840 |
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)}" |
|
841 |
using I by auto |
|
842 |
qed fact+ |
|
843 |
||
844 |
lemma prod_algebra_cong: |
|
845 |
assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))" |
|
846 |
shows "prod_algebra I M = prod_algebra J N" |
|
847 |
proof - |
|
848 |
have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)" |
|
849 |
using sets_eq_imp_space_eq[OF sets] by auto |
|
850 |
with sets show ?thesis unfolding `I = J` |
|
851 |
by (intro antisym prod_algebra_mono) auto |
|
852 |
qed |
|
853 |
||
854 |
lemma space_in_prod_algebra: |
|
855 |
"(\<Pi>\<^isub>E i\<in>I. space (M i)) \<in> prod_algebra I M" |
|
856 |
proof cases |
|
857 |
assume "I = {}" then show ?thesis |
|
858 |
by (auto simp add: prod_algebra_def image_iff prod_emb_def) |
|
859 |
next |
|
860 |
assume "I \<noteq> {}" |
|
861 |
then obtain i where "i \<in> I" by auto |
|
862 |
then have "(\<Pi>\<^isub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i))" |
|
863 |
by (auto simp: prod_emb_def Pi_iff) |
|
864 |
also have "\<dots> \<in> prod_algebra I M" |
|
865 |
using `i \<in> I` by (intro prod_algebraI) auto |
|
866 |
finally show ?thesis . |
|
867 |
qed |
|
868 |
||
869 |
lemma (in prob_space) indep_vars_iff_distr_eq_PiM: |
|
870 |
fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b" |
|
871 |
assumes "I \<noteq> {}" |
|
42988 | 872 |
assumes rv: "\<And>i. random_variable (M' i) (X i)" |
42989 | 873 |
shows "indep_vars M' X I \<longleftrightarrow> |
47694 | 874 |
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 | 875 |
proof - |
47694 | 876 |
let ?P = "\<Pi>\<^isub>M i\<in>I. M' i" |
877 |
let ?X = "\<lambda>x. \<lambda>i\<in>I. X i x" |
|
878 |
let ?D = "distr M ?P ?X" |
|
879 |
have X: "random_variable ?P ?X" by (intro measurable_restrict rv) |
|
880 |
interpret D: prob_space ?D by (intro prob_space_distr X) |
|
42988 | 881 |
|
47694 | 882 |
let ?D' = "\<lambda>i. distr M (M' i) (X i)" |
883 |
let ?P' = "\<Pi>\<^isub>M i\<in>I. distr M (M' i) (X i)" |
|
884 |
interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv) |
|
885 |
interpret P: product_prob_space ?D' I .. |
|
886 |
||
42988 | 887 |
show ?thesis |
47694 | 888 |
proof |
42989 | 889 |
assume "indep_vars M' X I" |
47694 | 890 |
show "?D = ?P'" |
891 |
proof (rule measure_eqI_generator_eq) |
|
892 |
show "Int_stable (prod_algebra I M')" |
|
893 |
by (rule Int_stable_prod_algebra) |
|
894 |
show "prod_algebra I M' \<subseteq> Pow (space ?P)" |
|
895 |
using prod_algebra_sets_into_space by (simp add: space_PiM) |
|
896 |
show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')" |
|
897 |
by (simp add: sets_PiM space_PiM) |
|
898 |
show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')" |
|
899 |
by (simp add: sets_PiM space_PiM cong: prod_algebra_cong) |
|
900 |
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
|
901 |
show "range ?A \<subseteq> prod_algebra I M'" "(\<Union>i. ?A i) = space (Pi\<^isub>M I M')" |
47694 | 902 |
by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong) |
903 |
{ fix i show "emeasure ?D (\<Pi>\<^isub>E i\<in>I. space (M' i)) \<noteq> \<infinity>" by auto } |
|
904 |
next |
|
905 |
fix E assume E: "E \<in> prod_algebra I M'" |
|
906 |
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
|
907 |
|
47694 | 908 |
from E have "E \<in> sets ?P" by (auto simp: sets_PiM) |
909 |
then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)" |
|
910 |
by (simp add: emeasure_distr X) |
|
911 |
also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)" |
|
912 |
using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm) |
|
913 |
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))" |
|
914 |
using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J] |
|
915 |
by (auto simp: emeasure_eq_measure setprod_ereal) |
|
916 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" |
|
917 |
using rv J by (simp add: emeasure_distr) |
|
918 |
also have "\<dots> = emeasure ?P' E" |
|
919 |
using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def) |
|
920 |
finally show "emeasure ?D E = emeasure ?P' E" . |
|
42988 | 921 |
qed |
922 |
next |
|
47694 | 923 |
assume "?D = ?P'" |
924 |
show "indep_vars M' X I" unfolding indep_vars_def |
|
925 |
proof (intro conjI indep_setsI ballI rv) |
|
926 |
fix i show "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} \<subseteq> events" |
|
927 |
by (auto intro!: sigma_sets_subset measurable_sets rv) |
|
42988 | 928 |
next |
47694 | 929 |
fix J Y' assume J: "J \<noteq> {}" "J \<subseteq> I" "finite J" |
930 |
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)}" |
|
931 |
have "\<forall>j\<in>J. \<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" |
|
42988 | 932 |
proof |
47694 | 933 |
fix j assume "j \<in> J" |
934 |
from Y'[rule_format, OF this] rv[of j] |
|
935 |
show "\<exists>Y. Y' j = X j -` Y \<inter> space M \<and> Y \<in> sets (M' j)" |
|
936 |
by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"]) |
|
937 |
(auto dest: measurable_space simp: sigma_sets_eq) |
|
42988 | 938 |
qed |
47694 | 939 |
from bchoice[OF this] obtain Y where |
940 |
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 |
|
941 |
let ?E = "prod_emb I M' J (Pi\<^isub>E J Y)" |
|
942 |
from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M" |
|
943 |
using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def Pi_iff split: split_if_asm) |
|
944 |
then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)" |
|
945 |
by simp |
|
946 |
also have "\<dots> = emeasure ?D ?E" |
|
947 |
using Y J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto |
|
948 |
also have "\<dots> = emeasure ?P' ?E" |
|
949 |
using `?D = ?P'` by simp |
|
950 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))" |
|
951 |
using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def) |
|
952 |
also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))" |
|
953 |
using rv J Y by (simp add: emeasure_distr) |
|
954 |
finally have "emeasure M (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. emeasure M (Y' i))" . |
|
955 |
then show "prob (\<Inter>j\<in>J. Y' j) = (\<Prod> i\<in>J. prob (Y' i))" |
|
956 |
by (auto simp: emeasure_eq_measure setprod_ereal) |
|
42988 | 957 |
qed |
958 |
qed |
|
42987 | 959 |
qed |
960 |
||
42989 | 961 |
lemma (in prob_space) indep_varD: |
962 |
assumes indep: "indep_var Ma A Mb B" |
|
963 |
assumes sets: "Xa \<in> sets Ma" "Xb \<in> sets Mb" |
|
964 |
shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = |
|
965 |
prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" |
|
966 |
proof - |
|
967 |
have "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = |
|
968 |
prob (\<Inter>i\<in>UNIV. (bool_case A B i -` bool_case Xa Xb i \<inter> space M))" |
|
969 |
by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool) |
|
970 |
also have "\<dots> = (\<Prod>i\<in>UNIV. prob (bool_case A B i -` bool_case Xa Xb i \<inter> space M))" |
|
971 |
using indep unfolding indep_var_def |
|
972 |
by (rule indep_varsD) (auto split: bool.split intro: sets) |
|
973 |
also have "\<dots> = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)" |
|
974 |
unfolding UNIV_bool by simp |
|
975 |
finally show ?thesis . |
|
976 |
qed |
|
977 |
||
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
978 |
lemma (in prob_space) |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
979 |
assumes "indep_var S X T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
980 |
shows indep_var_rv1: "random_variable S X" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
981 |
and indep_var_rv2: "random_variable T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
982 |
proof - |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
983 |
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
|
984 |
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
|
985 |
then show "random_variable S X" "random_variable T Y" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
986 |
unfolding UNIV_bool by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
987 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
988 |
|
47694 | 989 |
lemma measurable_bool_case[simp, intro]: |
990 |
"(\<lambda>(x, y). bool_case x y) \<in> measurable (M \<Otimes>\<^isub>M N) (Pi\<^isub>M UNIV (bool_case M N))" |
|
991 |
(is "?f \<in> measurable ?B ?P") |
|
992 |
proof (rule measurable_PiM_single) |
|
993 |
show "?f \<in> space ?B \<rightarrow> (\<Pi>\<^isub>E i\<in>UNIV. space (bool_case M N i))" |
|
994 |
by (auto simp: space_pair_measure extensional_def split: bool.split) |
|
995 |
fix i A assume "A \<in> sets (case i of True \<Rightarrow> M | False \<Rightarrow> N)" |
|
996 |
moreover then have "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A} |
|
997 |
= (case i of True \<Rightarrow> A \<times> space N | False \<Rightarrow> space M \<times> A)" |
|
998 |
by (auto simp: space_pair_measure split: bool.split dest!: sets_into_space) |
|
999 |
ultimately show "{\<omega> \<in> space (M \<Otimes>\<^isub>M N). prod_case bool_case \<omega> i \<in> A} \<in> sets ?B" |
|
1000 |
by (auto split: bool.split) |
|
1001 |
qed |
|
1002 |
||
1003 |
lemma borel_measurable_indicator': |
|
1004 |
"A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M" |
|
1005 |
using measurable_comp[OF _ borel_measurable_indicator, of f M N A] by (auto simp add: comp_def) |
|
1006 |
||
1007 |
lemma (in product_sigma_finite) distr_component: |
|
1008 |
"distr (M i) (Pi\<^isub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^isub>M {i} M" (is "?D = ?P") |
|
1009 |
proof (intro measure_eqI[symmetric]) |
|
1010 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
|
1011 |
||
1012 |
have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x" |
|
1013 |
by (auto simp: extensional_def restrict_def) |
|
1014 |
||
1015 |
fix A assume A: "A \<in> sets ?P" |
|
1016 |
then have "emeasure ?P A = (\<integral>\<^isup>+x. indicator A x \<partial>?P)" |
|
1017 |
by simp |
|
1018 |
also have "\<dots> = (\<integral>\<^isup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) x \<partial>M i)" |
|
1019 |
apply (subst product_positive_integral_singleton[symmetric]) |
|
1020 |
apply (force intro!: measurable_restrict measurable_sets A) |
|
1021 |
apply (auto intro!: positive_integral_cong simp: space_PiM indicator_def simp: eq) |
|
1022 |
done |
|
1023 |
also have "\<dots> = emeasure (M i) ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i))" |
|
1024 |
by (force intro!: measurable_restrict measurable_sets A positive_integral_indicator) |
|
1025 |
also have "\<dots> = emeasure ?D A" |
|
1026 |
using A by (auto intro!: emeasure_distr[symmetric] measurable_restrict) |
|
1027 |
finally show "emeasure (Pi\<^isub>M {i} M) A = emeasure ?D A" . |
|
1028 |
qed simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1029 |
|
47694 | 1030 |
lemma pair_measure_eqI: |
1031 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
1032 |
assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M" |
|
1033 |
assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)" |
|
1034 |
shows "M1 \<Otimes>\<^isub>M M2 = M" |
|
1035 |
proof - |
|
1036 |
interpret M1: sigma_finite_measure M1 by fact |
|
1037 |
interpret M2: sigma_finite_measure M2 by fact |
|
1038 |
interpret pair_sigma_finite M1 M2 by default |
|
1039 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
|
1040 |
let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
|
1041 |
let ?P = "M1 \<Otimes>\<^isub>M M2" |
|
1042 |
show ?thesis |
|
1043 |
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
|
1044 |
show "?E \<subseteq> Pow (space ?P)" |
|
1045 |
using space_closed[of M1] space_closed[of M2] by (auto simp: space_pair_measure) |
|
1046 |
show "sets ?P = sigma_sets (space ?P) ?E" |
|
1047 |
by (simp add: sets_pair_measure space_pair_measure) |
|
1048 |
then show "sets M = sigma_sets (space ?P) ?E" |
|
1049 |
using sets[symmetric] by simp |
|
1050 |
next |
|
49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49781
diff
changeset
|
1051 |
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
47694 | 1052 |
using F by (auto simp: space_pair_measure) |
1053 |
next |
|
1054 |
fix X assume "X \<in> ?E" |
|
1055 |
then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto |
|
1056 |
then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" |
|
49776 | 1057 |
by (simp add: M2.emeasure_pair_measure_Times) |
47694 | 1058 |
also have "\<dots> = emeasure M (A \<times> B)" |
1059 |
using A B emeasure by auto |
|
1060 |
finally show "emeasure ?P X = emeasure M X" |
|
1061 |
by simp |
|
1062 |
qed |
|
1063 |
qed |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1064 |
|
47694 | 1065 |
lemma pair_measure_eq_distr_PiM: |
1066 |
fixes M1 :: "'a measure" and M2 :: "'a measure" |
|
1067 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
1068 |
shows "(M1 \<Otimes>\<^isub>M M2) = distr (Pi\<^isub>M UNIV (bool_case M1 M2)) (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. (x True, x False))" |
|
1069 |
(is "?P = ?D") |
|
1070 |
proof (rule pair_measure_eqI[OF assms]) |
|
1071 |
interpret B: product_sigma_finite "bool_case M1 M2" |
|
1072 |
unfolding product_sigma_finite_def using assms by (auto split: bool.split) |
|
1073 |
let ?B = "Pi\<^isub>M UNIV (bool_case M1 M2)" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1074 |
|
47694 | 1075 |
have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)" |
1076 |
by auto |
|
1077 |
fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2" |
|
1078 |
have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (bool_case M1 M2 i) (bool_case A B i))" |
|
1079 |
by (simp add: UNIV_bool ac_simps) |
|
1080 |
also have "\<dots> = emeasure ?B (Pi\<^isub>E UNIV (bool_case A B))" |
|
1081 |
using A B by (subst B.emeasure_PiM) (auto split: bool.split) |
|
1082 |
also have "Pi\<^isub>E UNIV (bool_case A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B" |
|
1083 |
using A[THEN sets_into_space] B[THEN sets_into_space] |
|
1084 |
by (auto simp: Pi_iff all_bool_eq space_PiM split: bool.split) |
|
1085 |
finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)" |
|
1086 |
using A B |
|
1087 |
measurable_component_singleton[of True UNIV "bool_case M1 M2"] |
|
1088 |
measurable_component_singleton[of False UNIV "bool_case M1 M2"] |
|
1089 |
by (subst emeasure_distr) (auto simp: measurable_pair_iff) |
|
1090 |
qed simp |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1091 |
|
47694 | 1092 |
lemma measurable_Pair: |
1093 |
assumes rvs: "X \<in> measurable M S" "Y \<in> measurable M T" |
|
1094 |
shows "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)" |
|
1095 |
proof - |
|
1096 |
have [simp]: "fst \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. X x)" "snd \<circ> (\<lambda>x. (X x, Y x)) = (\<lambda>x. Y x)" |
|
1097 |
by auto |
|
1098 |
show " (\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)" |
|
1099 |
by (auto simp: measurable_pair_iff rvs) |
|
1100 |
qed |
|
1101 |
||
1102 |
lemma (in prob_space) indep_var_distribution_eq: |
|
1103 |
"indep_var S X T Y \<longleftrightarrow> random_variable S X \<and> random_variable T Y \<and> |
|
1104 |
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") |
|
1105 |
proof safe |
|
1106 |
assume "indep_var S X T Y" |
|
1107 |
then show rvs: "random_variable S X" "random_variable T Y" |
|
1108 |
by (blast dest: indep_var_rv1 indep_var_rv2)+ |
|
1109 |
then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
1110 |
by (rule measurable_Pair) |
|
1111 |
||
1112 |
interpret X: prob_space ?S by (rule prob_space_distr) fact |
|
1113 |
interpret Y: prob_space ?T by (rule prob_space_distr) fact |
|
1114 |
interpret XY: pair_prob_space ?S ?T .. |
|
1115 |
show "?S \<Otimes>\<^isub>M ?T = ?J" |
|
1116 |
proof (rule pair_measure_eqI) |
|
1117 |
show "sigma_finite_measure ?S" .. |
|
1118 |
show "sigma_finite_measure ?T" .. |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1119 |
|
47694 | 1120 |
fix A B assume A: "A \<in> sets ?S" and B: "B \<in> sets ?T" |
1121 |
have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)" |
|
1122 |
using A B by (intro emeasure_distr[OF XY]) auto |
|
1123 |
also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)" |
|
1124 |
using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure) |
|
1125 |
also have "\<dots> = emeasure ?S A * emeasure ?T B" |
|
1126 |
using rvs A B by (simp add: emeasure_distr) |
|
1127 |
finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp |
|
1128 |
qed simp |
|
1129 |
next |
|
1130 |
assume rvs: "random_variable S X" "random_variable T Y" |
|
1131 |
then have XY: "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" |
|
1132 |
by (rule measurable_Pair) |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1133 |
|
47694 | 1134 |
let ?S = "distr M S X" and ?T = "distr M T Y" |
1135 |
interpret X: prob_space ?S by (rule prob_space_distr) fact |
|
1136 |
interpret Y: prob_space ?T by (rule prob_space_distr) fact |
|
1137 |
interpret XY: pair_prob_space ?S ?T .. |
|
1138 |
||
1139 |
assume "?S \<Otimes>\<^isub>M ?T = ?J" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1140 |
|
47694 | 1141 |
{ fix S and X |
1142 |
have "Int_stable {X -` A \<inter> space M |A. A \<in> sets S}" |
|
1143 |
proof (safe intro!: Int_stableI) |
|
1144 |
fix A B assume "A \<in> sets S" "B \<in> sets S" |
|
1145 |
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" |
|
1146 |
by (intro exI[of _ "A \<inter> B"]) auto |
|
1147 |
qed } |
|
1148 |
note Int_stable = this |
|
1149 |
||
1150 |
show "indep_var S X T Y" unfolding indep_var_eq |
|
1151 |
proof (intro conjI indep_set_sigma_sets Int_stable rvs) |
|
1152 |
show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}" |
|
1153 |
proof (safe intro!: indep_setI) |
|
1154 |
{ fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M" |
|
1155 |
using `X \<in> measurable M S` by (auto intro: measurable_sets) } |
|
1156 |
{ fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M" |
|
1157 |
using `Y \<in> measurable M T` by (auto intro: measurable_sets) } |
|
1158 |
next |
|
1159 |
fix A B assume ab: "A \<in> sets S" "B \<in> sets T" |
|
1160 |
then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)" |
|
1161 |
using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"]) |
|
1162 |
also have "\<dots> = emeasure (?S \<Otimes>\<^isub>M ?T) (A \<times> B)" |
|
1163 |
unfolding `?S \<Otimes>\<^isub>M ?T = ?J` .. |
|
1164 |
also have "\<dots> = emeasure ?S A * emeasure ?T B" |
|
49776 | 1165 |
using ab by (simp add: Y.emeasure_pair_measure_Times) |
47694 | 1166 |
finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) = |
1167 |
prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)" |
|
1168 |
using rvs ab by (simp add: emeasure_eq_measure emeasure_distr) |
|
1169 |
qed |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1170 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42989
diff
changeset
|
1171 |
qed |
42989 | 1172 |
|
42861
16375b493b64
Add formalization of probabilistic independence for families of sets
hoelzl
parents:
diff
changeset
|
1173 |
end |