src/HOL/Probability/Independent_Family.thy
author hoelzl
Tue May 17 11:47:36 2011 +0200 (2011-05-17)
changeset 42861 16375b493b64
child 42981 fe7f5a26e4c6
permissions -rw-r--r--
Add formalization of probabilistic independence for families of sets
     1 (*  Title:      HOL/Probability/Independent_Family.thy
     2     Author:     Johannes Hölzl, TU München
     3 *)
     4 
     5 header {* Independent families of events, event sets, and random variables *}
     6 
     7 theory Independent_Family
     8   imports Probability_Measure
     9 begin
    10 
    11 definition (in prob_space)
    12   "indep_events A I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
    13 
    14 definition (in prob_space)
    15   "indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> events) \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow>
    16     (\<forall>A\<in>(\<Pi> j\<in>J. F j). prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))))"
    17 
    18 definition (in prob_space)
    19   "indep_sets2 A B \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
    20 
    21 definition (in prob_space)
    22   "indep_rv M' X I \<longleftrightarrow>
    23     (\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
    24     indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
    25 
    26 lemma (in prob_space) indep_sets_finite_index_sets:
    27   "indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
    28 proof (intro iffI allI impI)
    29   assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
    30   show "indep_sets F I" unfolding indep_sets_def
    31   proof (intro conjI ballI allI impI)
    32     fix i assume "i \<in> I"
    33     with *[THEN spec, of "{i}"] show "F i \<subseteq> events"
    34       by (auto simp: indep_sets_def)
    35   qed (insert *, auto simp: indep_sets_def)
    36 qed (auto simp: indep_sets_def)
    37 
    38 lemma (in prob_space) indep_sets_mono_index:
    39   "J \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> indep_sets F J"
    40   unfolding indep_sets_def by auto
    41 
    42 lemma (in prob_space) indep_sets_mono_sets:
    43   assumes indep: "indep_sets F I"
    44   assumes mono: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
    45   shows "indep_sets G I"
    46 proof -
    47   have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
    48     using mono by auto
    49   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)"
    50     using mono by (auto simp: Pi_iff)
    51   ultimately show ?thesis
    52     using indep by (auto simp: indep_sets_def)
    53 qed
    54 
    55 lemma (in prob_space) indep_setsI:
    56   assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
    57     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))"
    58   shows "indep_sets F I"
    59   using assms unfolding indep_sets_def by (auto simp: Pi_iff)
    60 
    61 lemma (in prob_space) indep_setsD:
    62   assumes "indep_sets F I" and "J \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
    63   shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
    64   using assms unfolding indep_sets_def by auto
    65 
    66 lemma dynkin_systemI':
    67   assumes 1: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
    68   assumes empty: "{} \<in> sets M"
    69   assumes Diff: "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
    70   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
    71           \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
    72   shows "dynkin_system M"
    73 proof -
    74   from Diff[OF empty] have "space M \<in> sets M" by auto
    75   from 1 this Diff 2 show ?thesis
    76     by (intro dynkin_systemI) auto
    77 qed
    78 
    79 lemma (in prob_space) indep_sets_dynkin:
    80   assumes indep: "indep_sets F I"
    81   shows "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) I"
    82     (is "indep_sets ?F I")
    83 proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
    84   fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
    85   with indep have "indep_sets F J"
    86     by (subst (asm) indep_sets_finite_index_sets) auto
    87   { fix J K assume "indep_sets F K"
    88     let "?G S i" = "if i \<in> S then ?F i else F i"
    89     assume "finite J" "J \<subseteq> K"
    90     then have "indep_sets (?G J) K"
    91     proof induct
    92       case (insert j J)
    93       moreover def G \<equiv> "?G J"
    94       ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
    95         by (auto simp: indep_sets_def)
    96       let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
    97       { fix X assume X: "X \<in> events"
    98         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)
    99           \<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
   100         have "indep_sets (G(j := {X})) K"
   101         proof (rule indep_setsI)
   102           fix i assume "i \<in> K" then show "(G(j:={X})) i \<subseteq> events"
   103             using G X by auto
   104         next
   105           fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
   106           show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   107           proof cases
   108             assume "j \<in> J"
   109             with J have "A j = X" by auto
   110             show ?thesis
   111             proof cases
   112               assume "J = {j}" then show ?thesis by simp
   113             next
   114               assume "J \<noteq> {j}"
   115               have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
   116                 using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   117               also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
   118               proof (rule indep)
   119                 show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
   120                   using J `J \<noteq> {j}` `j \<in> J` by auto
   121                 show "\<forall>i\<in>J - {j}. A i \<in> G i"
   122                   using J by auto
   123               qed
   124               also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
   125                 using `A j = X` by simp
   126               also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
   127                 unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
   128                 using `j \<in> J` by (simp add: insert_absorb)
   129               finally show ?thesis .
   130             qed
   131           next
   132             assume "j \<notin> J"
   133             with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   134             with J show ?thesis
   135               by (intro indep_setsD[OF G(1)]) auto
   136           qed
   137         qed }
   138       note indep_sets_insert = this
   139       have "dynkin_system \<lparr> space = space M, sets = ?D \<rparr>"
   140       proof (rule dynkin_systemI', simp_all, safe)
   141         show "indep_sets (G(j := {{}})) K"
   142           by (rule indep_sets_insert) auto
   143       next
   144         fix X assume X: "X \<in> events" and G': "indep_sets (G(j := {X})) K"
   145         show "indep_sets (G(j := {space M - X})) K"
   146         proof (rule indep_sets_insert)
   147           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"
   148           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   149             using G by auto
   150           have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   151               prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
   152             using A_sets sets_into_space X `J \<noteq> {}`
   153             by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   154           also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
   155             using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
   156             by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
   157           finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   158               prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
   159           moreover {
   160             have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   161               using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
   162             then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
   163               using prob_space by simp }
   164           moreover {
   165             have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
   166               using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
   167             then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
   168               using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
   169           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))"
   170             by (simp add: field_simps)
   171           also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
   172             using X A by (simp add: finite_measure_compl)
   173           finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
   174         qed (insert X, auto)
   175       next
   176         fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
   177         then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
   178         show "indep_sets (G(j := {\<Union>k. F k})) K"
   179         proof (rule indep_sets_insert)
   180           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"
   181           then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   182             using G by auto
   183           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))"
   184             using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   185           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))"
   186           proof (rule finite_measure_UNION)
   187             show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
   188               using disj by (rule disjoint_family_on_bisimulation) auto
   189             show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
   190               using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
   191           qed
   192           moreover { fix k
   193             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))"
   194               by (subst indep_setsD[OF F(2)]) (auto intro!: setprod_cong split: split_if_asm)
   195             also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
   196               using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
   197             finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
   198           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)))"
   199             by simp
   200           moreover
   201           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))"
   202             using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
   203           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)))"
   204             using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
   205           ultimately
   206           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))"
   207             by (auto dest!: sums_unique)
   208         qed (insert F, auto)
   209       qed (insert sets_into_space, auto)
   210       then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>
   211         sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"
   212       proof (rule dynkin_system.dynkin_subset, simp_all, safe)
   213         fix X assume "X \<in> G j"
   214         then show "X \<in> events" using G `j \<in> K` by auto
   215         from `indep_sets G K`
   216         show "indep_sets (G(j := {X})) K"
   217           by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
   218       qed
   219       have "indep_sets (G(j:=?D)) K"
   220       proof (rule indep_setsI)
   221         fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
   222           using G(2) by auto
   223       next
   224         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"
   225         show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   226         proof cases
   227           assume "j \<in> J"
   228           with A have indep: "indep_sets (G(j := {A j})) K" by auto
   229           from J A show ?thesis
   230             by (intro indep_setsD[OF indep]) auto
   231         next
   232           assume "j \<notin> J"
   233           with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
   234           with J show ?thesis
   235             by (intro indep_setsD[OF G(1)]) auto
   236         qed
   237       qed
   238       then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K"
   239         by (rule indep_sets_mono_sets) (insert mono, auto)
   240       then show ?case
   241         by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
   242     qed (insert `indep_sets F K`, simp) }
   243   from this[OF `indep_sets F J` `finite J` subset_refl]
   244   show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J"
   245     by (rule indep_sets_mono_sets) auto
   246 qed
   247 
   248 lemma (in prob_space) indep_sets_sigma:
   249   assumes indep: "indep_sets F I"
   250   assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
   251   shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) I"
   252 proof -
   253   from indep_sets_dynkin[OF indep]
   254   show ?thesis
   255   proof (rule indep_sets_mono_sets, subst sigma_eq_dynkin, simp_all add: stable)
   256     fix i assume "i \<in> I"
   257     with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
   258     with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
   259   qed
   260 qed
   261 
   262 lemma (in prob_space) indep_sets_sigma_sets:
   263   assumes "indep_sets F I"
   264   assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
   265   shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
   266   using indep_sets_sigma[OF assms] by (simp add: sets_sigma)
   267 
   268 lemma (in prob_space) indep_sets2_eq:
   269   "indep_sets2 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)"
   270   unfolding indep_sets2_def
   271 proof (intro iffI ballI conjI)
   272   assume indep: "indep_sets (bool_case A B) UNIV"
   273   { fix a b assume "a \<in> A" "b \<in> B"
   274     with indep_setsD[OF indep, of UNIV "bool_case a b"]
   275     show "prob (a \<inter> b) = prob a * prob b"
   276       unfolding UNIV_bool by (simp add: ac_simps) }
   277   from indep show "A \<subseteq> events" "B \<subseteq> events"
   278     unfolding indep_sets_def UNIV_bool by auto
   279 next
   280   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)"
   281   show "indep_sets (bool_case A B) UNIV"
   282   proof (rule indep_setsI)
   283     fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
   284       using * by (auto split: bool.split)
   285   next
   286     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)"
   287     then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
   288       by (auto simp: UNIV_bool)
   289     then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>J. prob (X j))"
   290       using X * by auto
   291   qed
   292 qed
   293 
   294 lemma (in prob_space) indep_sets2_sigma_sets:
   295   assumes "indep_sets2 A B"
   296   assumes A: "Int_stable \<lparr> space = space M, sets = A \<rparr>"
   297   assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"
   298   shows "indep_sets2 (sigma_sets (space M) A) (sigma_sets (space M) B)"
   299 proof -
   300   have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
   301   proof (rule indep_sets_sigma_sets)
   302     show "indep_sets (bool_case A B) UNIV"
   303       by (rule `indep_sets2 A B`[unfolded indep_sets2_def])
   304     fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"
   305       using A B by (cases i) auto
   306   qed
   307   then show ?thesis
   308     unfolding indep_sets2_def
   309     by (rule indep_sets_mono_sets) (auto split: bool.split)
   310 qed
   311 
   312 end