src/HOL/Probability/Probability_Mass_Function.thy
author hoelzl
Tue Oct 07 10:34:24 2014 +0200 (2014-10-07)
changeset 58606 9c66f7c541fb
parent 58587 5484f6079bcd
child 58730 b3fd0628f849
permissions -rw-r--r--
add Giry monad
     1 (*  Title:      HOL/Probability/Probability_Mass_Function.thy
     2     Author:     Johannes Hölzl, TU München *)
     3 
     4 theory Probability_Mass_Function
     5   imports Probability_Measure
     6 begin
     7 
     8 lemma (in prob_space) countable_support:
     9   "countable {x. measure M {x} \<noteq> 0}"
    10 proof -
    11   let ?m = "\<lambda>x. measure M {x}"
    12   have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. inverse (real (Suc n)) < ?m x})"
    13     by (auto intro!: measure_nonneg reals_Archimedean order_le_neq_trans)
    14   have **: "\<And>n. finite {x. inverse (Suc n) < ?m x}"
    15   proof (rule ccontr)
    16     fix n assume "infinite {x. inverse (Suc n) < ?m x}" (is "infinite ?X")
    17     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
    18       by (metis infinite_arbitrarily_large)
    19     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> 1 / Suc n \<le> ?m x" 
    20       by (auto simp: inverse_eq_divide)
    21     { fix x assume "x \<in> X"
    22       from *[OF this] have "?m x \<noteq> 0" by auto
    23       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
    24     note singleton_sets = this
    25     have "1 < (\<Sum>x\<in>X. 1 / Suc n)"
    26       by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc)
    27     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
    28       by (rule setsum_mono) fact
    29     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
    30       using singleton_sets `finite X`
    31       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
    32     finally show False
    33       using prob_le_1[of "\<Union>x\<in>X. {x}"] by arith
    34   qed
    35   show ?thesis
    36     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
    37 qed
    38 
    39 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
    40   morphisms measure_pmf Abs_pmf
    41   by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
    42      (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
    43 
    44 declare [[coercion measure_pmf]]
    45 
    46 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
    47   using pmf.measure_pmf[of p] by auto
    48 
    49 interpretation measure_pmf!: prob_space "measure_pmf M" for M
    50   by (rule prob_space_measure_pmf)
    51 
    52 locale pmf_as_measure
    53 begin
    54 
    55 setup_lifting type_definition_pmf
    56 
    57 end
    58 
    59 context
    60 begin
    61 
    62 interpretation pmf_as_measure .
    63 
    64 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
    65 
    66 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
    67 
    68 lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
    69   "\<lambda>f M. distr M (count_space UNIV) f"
    70 proof safe
    71   fix M and f :: "'a \<Rightarrow> 'b"
    72   let ?D = "distr M (count_space UNIV) f"
    73   assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
    74   interpret prob_space M by fact
    75   from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
    76   proof eventually_elim
    77     fix x
    78     have "measure M {x} \<le> measure M (f -` {f x})"
    79       by (intro finite_measure_mono) auto
    80     then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
    81       using measure_nonneg[of M "{x}"] by auto
    82   qed
    83   then show "AE x in ?D. measure ?D {x} \<noteq> 0"
    84     by (simp add: AE_distr_iff measure_distr measurable_def)
    85 qed (auto simp: measurable_def prob_space.prob_space_distr)
    86 
    87 declare [[coercion set_pmf]]
    88 
    89 lemma countable_set_pmf: "countable (set_pmf p)"
    90   by transfer (metis prob_space.countable_support)
    91 
    92 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
    93   by transfer metis
    94 
    95 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
    96   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
    97 
    98 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
    99   by (auto simp: measurable_def)
   100 
   101 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
   102   by (intro measurable_cong_sets) simp_all
   103 
   104 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
   105   by transfer (simp add: less_le measure_nonneg)
   106 
   107 lemma pmf_nonneg: "0 \<le> pmf p x"
   108   by transfer (simp add: measure_nonneg)
   109 
   110 lemma emeasure_pmf_single:
   111   fixes M :: "'a pmf"
   112   shows "emeasure M {x} = pmf M x"
   113   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
   114 
   115 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
   116   by transfer simp
   117 
   118 lemma emeasure_pmf_single_eq_zero_iff:
   119   fixes M :: "'a pmf"
   120   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
   121   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
   122 
   123 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
   124 proof -
   125   { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
   126     with P have "AE x in M. x \<noteq> y"
   127       by auto
   128     with y have False
   129       by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
   130   then show ?thesis
   131     using AE_measure_pmf[of M] by auto
   132 qed
   133 
   134 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
   135 proof (transfer, elim conjE)
   136   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
   137   assume "prob_space M" then interpret prob_space M .
   138   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
   139   proof (rule measure_eqI)
   140     fix A :: "'a set"
   141     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = 
   142       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
   143       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
   144     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
   145       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
   146     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
   147       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
   148          (auto simp: disjoint_family_on_def)
   149     also have "\<dots> = emeasure M A"
   150       using ae by (intro emeasure_eq_AE) auto
   151     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
   152       using emeasure_space_1 by (simp add: emeasure_density)
   153   qed simp
   154 qed
   155 
   156 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
   157   using AE_measure_pmf[of M] by (intro notI) simp
   158 
   159 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
   160   by transfer simp
   161 
   162 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
   163 proof -
   164   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
   165     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
   166   then show ?thesis
   167     using measure_pmf.emeasure_space_1 by simp
   168 qed
   169 
   170 lemma map_pmf_id[simp]: "map_pmf id = id"
   171   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
   172 
   173 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
   174   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 
   175 
   176 lemma map_pmf_cong:
   177   assumes "p = q"
   178   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
   179   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
   180   by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
   181 
   182 lemma pmf_set_map: 
   183   fixes f :: "'a \<Rightarrow> 'b"
   184   shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
   185 proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
   186   fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
   187   assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
   188   interpret prob_space M by fact
   189   show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
   190   proof safe
   191     fix x assume "measure M (f -` {x}) \<noteq> 0"
   192     moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
   193       using ae by (intro finite_measure_eq_AE) auto
   194     ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
   195       by (metis measure_empty)
   196     then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
   197       by auto
   198   next
   199     fix x assume "measure M {x} \<noteq> 0"
   200     then have "0 < measure M {x}"
   201       using measure_nonneg[of M "{x}"] by auto
   202     also have "measure M {x} \<le> measure M (f -` {f x})"
   203       by (intro finite_measure_mono) auto
   204     finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
   205       by simp
   206   qed
   207 qed
   208 
   209 context
   210   fixes f :: "'a \<Rightarrow> real"
   211   assumes nonneg: "\<And>x. 0 \<le> f x"
   212   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
   213 begin
   214 
   215 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
   216 proof (intro conjI)
   217   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
   218     by (simp split: split_indicator)
   219   show "AE x in density (count_space UNIV) (ereal \<circ> f).
   220     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
   221     by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator)
   222   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
   223     by default (simp add: emeasure_density prob)
   224 qed simp
   225 
   226 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
   227 proof transfer
   228   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
   229     by (simp split: split_indicator)
   230   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
   231     by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg)
   232 qed
   233 
   234 end
   235 
   236 lemma embed_pmf_transfer:
   237   "rel_fun (eq_onp (\<lambda>f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
   238   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
   239 
   240 lemma td_pmf_embed_pmf:
   241   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
   242   unfolding type_definition_def
   243 proof safe
   244   fix p :: "'a pmf"
   245   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
   246     using measure_pmf.emeasure_space_1[of p] by simp
   247   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
   248     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
   249 
   250   show "embed_pmf (pmf p) = p"
   251     by (intro measure_pmf_inject[THEN iffD1])
   252        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
   253 next
   254   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
   255   then show "pmf (embed_pmf f) = f"
   256     by (auto intro!: pmf_embed_pmf)
   257 qed (rule pmf_nonneg)
   258 
   259 end
   260 
   261 locale pmf_as_function
   262 begin
   263 
   264 setup_lifting td_pmf_embed_pmf
   265 
   266 end 
   267 
   268 (*
   269 
   270 definition
   271   "rel_pmf P d1 d2 \<longleftrightarrow> (\<exists>p3. (\<forall>(x, y) \<in> set_pmf p3. P x y) \<and> map_pmf fst p3 = d1 \<and> map_pmf snd p3 = d2)"
   272 
   273 lift_definition pmf_join :: "real \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf" is
   274   "\<lambda>p M1 M2. density (count_space UNIV) (\<lambda>x. p * measure M1 {x} + (1 - p) * measure M2 {x})"
   275 sorry
   276 
   277 lift_definition pmf_single :: "'a \<Rightarrow> 'a pmf" is
   278   "\<lambda>x. uniform_measure (count_space UNIV) {x}"
   279 sorry
   280 
   281 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel
   282 proof -
   283   show "map_pmf id = id" by (rule map_pmf_id)
   284   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) 
   285   show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
   286     by (intro map_pmg_cong refl)
   287 
   288   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
   289     by (rule pmf_set_map)
   290 
   291   { fix p :: "'s pmf"
   292     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
   293       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
   294          (auto intro: countable_set_pmf inj_on_to_nat_on)
   295     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
   296       by (metis Field_natLeq card_of_least natLeq_Well_order)
   297     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
   298 
   299   show "\<And>R. pmf_rel R =
   300          (BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
   301          BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
   302      by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def)
   303 
   304   { let ?f = "map_pmf fst" and ?s = "map_pmf snd"
   305     fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and A assume "\<And>x y. (x, y) \<in> set_pmf A \<Longrightarrow> R x y"
   306     fix S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" and B assume "\<And>y z. (y, z) \<in> set_pmf B \<Longrightarrow> S y z"
   307     assume "?f B = ?s A"
   308     have "\<exists>C. (\<forall>(x, z)\<in>set_pmf C. \<exists>y. R x y \<and> S y z) \<and> ?f C = ?f A \<and> ?s C = ?s B"
   309       sorry }
   310 oops
   311   then show "\<And>R::'a \<Rightarrow> 'b \<Rightarrow> bool. \<And>S::'b \<Rightarrow> 'c \<Rightarrow> bool. pmf_rel R OO pmf_rel S \<le> pmf_rel (R OO S)"
   312       by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def)
   313 qed (fact natLeq_card_order natLeq_cinfinite)+
   314 
   315 notepad
   316 begin
   317   fix x y :: "nat \<Rightarrow> real"
   318   def IJz \<equiv> "rec_nat ((0, 0), \<lambda>_. 0) (\<lambda>n ((I, J), z).
   319     let a = x I - (\<Sum>j<J. z (I, j)) ; b = y J - (\<Sum>i<I. z (i, J)) in
   320       ((if a \<le> b then I + 1 else I, if b \<le> a then J + 1 else J), z((I, J) := min a b)))"
   321   def I == "fst \<circ> fst \<circ> IJz" def J == "snd \<circ> fst \<circ> IJz" def z == "snd \<circ> IJz"
   322   let ?a = "\<lambda>n. x (I n) - (\<Sum>j<J n. z n (I n, j))" and ?b = "\<lambda>n. y (J n) - (\<Sum>i<I n. z n (i, J n))"
   323   have IJz_0[simp]: "\<And>p. z 0 p = 0" "I 0 = 0" "J 0 = 0"
   324     by (simp_all add: I_def J_def z_def IJz_def)
   325   have z_Suc[simp]: "\<And>n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))"
   326     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
   327   have I_Suc[simp]: "\<And>n. I (Suc n) = (if ?a n \<le> ?b n then I n + 1 else I n)"
   328     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
   329   have J_Suc[simp]: "\<And>n. J (Suc n) = (if ?b n \<le> ?a n then J n + 1 else J n)"
   330     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
   331   
   332   { fix N have "\<And>p. z N p \<noteq> 0 \<Longrightarrow> \<exists>n<N. p = (I n, J n)"
   333       by (induct N) (auto simp add: less_Suc_eq split: split_if_asm) }
   334   
   335   { fix i n assume "i < I n"
   336     then have "(\<Sum>j. z n (i, j)) = x i" 
   337     oops
   338 *)
   339 
   340 end
   341