author | hoelzl |
Tue, 21 Oct 2014 17:00:42 +0200 | |
changeset 58730 | b3fd0628f849 |
parent 58606 | 9c66f7c541fb |
child 59000 | 6eb0725503fc |
permissions | -rw-r--r-- |
58606 | 1 |
(* Title: HOL/Probability/Probability_Mass_Function.thy |
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Author: Johannes Hölzl, TU München *) |
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theory Probability_Mass_Function |
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imports Probability_Measure |
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begin |
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|
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lemma (in prob_space) countable_support: |
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"countable {x. measure M {x} \<noteq> 0}" |
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proof - |
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let ?m = "\<lambda>x. measure M {x}" |
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have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. inverse (real (Suc n)) < ?m x})" |
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by (auto intro!: measure_nonneg reals_Archimedean order_le_neq_trans) |
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have **: "\<And>n. finite {x. inverse (Suc n) < ?m x}" |
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proof (rule ccontr) |
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fix n assume "infinite {x. inverse (Suc n) < ?m x}" (is "infinite ?X") |
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then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X" |
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by (metis infinite_arbitrarily_large) |
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from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> 1 / Suc n \<le> ?m x" |
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by (auto simp: inverse_eq_divide) |
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{ fix x assume "x \<in> X" |
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from *[OF this] have "?m x \<noteq> 0" by auto |
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then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) } |
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note singleton_sets = this |
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have "1 < (\<Sum>x\<in>X. 1 / Suc n)" |
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by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc) |
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also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)" |
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by (rule setsum_mono) fact |
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also have "\<dots> = measure M (\<Union>x\<in>X. {x})" |
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using singleton_sets `finite X` |
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by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def) |
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finally show False |
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using prob_le_1[of "\<Union>x\<in>X. {x}"] by arith |
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qed |
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show ?thesis |
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unfolding * by (intro countable_UN countableI_type countable_finite[OF **]) |
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qed |
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typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}" |
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morphisms measure_pmf Abs_pmf |
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by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"]) |
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(auto intro!: prob_space_uniform_measure AE_uniform_measureI) |
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declare [[coercion measure_pmf]] |
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lemma prob_space_measure_pmf: "prob_space (measure_pmf p)" |
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using pmf.measure_pmf[of p] by auto |
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interpretation measure_pmf!: prob_space "measure_pmf M" for M |
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by (rule prob_space_measure_pmf) |
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locale pmf_as_measure |
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begin |
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|
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setup_lifting type_definition_pmf |
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end |
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|
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context |
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begin |
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|
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interpretation pmf_as_measure . |
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lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" . |
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lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" . |
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lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is |
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"\<lambda>f M. distr M (count_space UNIV) f" |
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proof safe |
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fix M and f :: "'a \<Rightarrow> 'b" |
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let ?D = "distr M (count_space UNIV) f" |
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assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
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interpret prob_space M by fact |
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from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0" |
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proof eventually_elim |
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fix x |
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have "measure M {x} \<le> measure M (f -` {f x})" |
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by (intro finite_measure_mono) auto |
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then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0" |
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using measure_nonneg[of M "{x}"] by auto |
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qed |
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83 |
then show "AE x in ?D. measure ?D {x} \<noteq> 0" |
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by (simp add: AE_distr_iff measure_distr measurable_def) |
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qed (auto simp: measurable_def prob_space.prob_space_distr) |
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|
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declare [[coercion set_pmf]] |
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lemma countable_set_pmf: "countable (set_pmf p)" |
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90 |
by transfer (metis prob_space.countable_support) |
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91 |
|
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lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" |
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93 |
by transfer metis |
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94 |
|
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lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV" |
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using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp |
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98 |
lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N" |
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by (auto simp: measurable_def) |
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100 |
|
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101 |
lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)" |
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102 |
by (intro measurable_cong_sets) simp_all |
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103 |
|
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104 |
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x" |
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105 |
by transfer (simp add: less_le measure_nonneg) |
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106 |
|
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107 |
lemma pmf_nonneg: "0 \<le> pmf p x" |
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108 |
by transfer (simp add: measure_nonneg) |
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109 |
|
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110 |
lemma emeasure_pmf_single: |
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111 |
fixes M :: "'a pmf" |
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|
112 |
shows "emeasure M {x} = pmf M x" |
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113 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
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114 |
|
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115 |
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M" |
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116 |
by transfer simp |
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117 |
|
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118 |
lemma emeasure_pmf_single_eq_zero_iff: |
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119 |
fixes M :: "'a pmf" |
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|
120 |
shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M" |
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121 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
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122 |
|
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123 |
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)" |
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124 |
proof - |
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125 |
{ fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y" |
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126 |
with P have "AE x in M. x \<noteq> y" |
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127 |
by auto |
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|
128 |
with y have False |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
129 |
by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) } |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
130 |
then show ?thesis |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
131 |
using AE_measure_pmf[of M] by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
132 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
133 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
134 |
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
135 |
proof (transfer, elim conjE) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
136 |
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
137 |
assume "prob_space M" then interpret prob_space M . |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
138 |
show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
139 |
proof (rule measure_eqI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
140 |
fix A :: "'a set" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
141 |
have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
142 |
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
143 |
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
144 |
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
145 |
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
146 |
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
147 |
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
148 |
(auto simp: disjoint_family_on_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
149 |
also have "\<dots> = emeasure M A" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
150 |
using ae by (intro emeasure_eq_AE) auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
151 |
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
152 |
using emeasure_space_1 by (simp add: emeasure_density) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
153 |
qed simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
154 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
155 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
156 |
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
157 |
using AE_measure_pmf[of M] by (intro notI) simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
158 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
159 |
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
160 |
by transfer simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
161 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
162 |
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
163 |
proof - |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
164 |
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
165 |
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
166 |
then show ?thesis |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
167 |
using measure_pmf.emeasure_space_1 by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
168 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
169 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
170 |
lemma map_pmf_id[simp]: "map_pmf id = id" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
171 |
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
172 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
173 |
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
174 |
by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
175 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
176 |
lemma map_pmf_cong: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
177 |
assumes "p = q" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
178 |
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
179 |
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
180 |
by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
181 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
182 |
lemma pmf_set_map: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
183 |
fixes f :: "'a \<Rightarrow> 'b" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
184 |
shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
185 |
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
186 |
fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
187 |
assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
188 |
interpret prob_space M by fact |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
189 |
show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
190 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
191 |
fix x assume "measure M (f -` {x}) \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
192 |
moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
193 |
using ae by (intro finite_measure_eq_AE) auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
194 |
ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
195 |
by (metis measure_empty) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
196 |
then show "x \<in> f ` {x. measure M {x} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
197 |
by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
198 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
199 |
fix x assume "measure M {x} \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
200 |
then have "0 < measure M {x}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
201 |
using measure_nonneg[of M "{x}"] by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
202 |
also have "measure M {x} \<le> measure M (f -` {f x})" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
203 |
by (intro finite_measure_mono) auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
204 |
finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
205 |
by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
206 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
207 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
208 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
209 |
context |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
210 |
fixes f :: "'a \<Rightarrow> real" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
211 |
assumes nonneg: "\<And>x. 0 \<le> f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
212 |
assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
213 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
214 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
215 |
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
216 |
proof (intro conjI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
217 |
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
218 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
219 |
show "AE x in density (count_space UNIV) (ereal \<circ> f). |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
220 |
measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
221 |
by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
222 |
show "prob_space (density (count_space UNIV) (ereal \<circ> f))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
223 |
by default (simp add: emeasure_density prob) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
224 |
qed simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
225 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
226 |
lemma pmf_embed_pmf: "pmf embed_pmf x = f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
227 |
proof transfer |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
228 |
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
229 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
230 |
fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
231 |
by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
232 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
233 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
234 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
235 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
236 |
lemma embed_pmf_transfer: |
58730 | 237 |
"rel_fun (eq_onp (\<lambda>f. (\<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" |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
238 |
by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
239 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
240 |
lemma td_pmf_embed_pmf: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
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}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
242 |
unfolding type_definition_def |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
243 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
244 |
fix p :: "'a pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
245 |
have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
246 |
using measure_pmf.emeasure_space_1[of p] by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
247 |
then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
248 |
by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
249 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
250 |
show "embed_pmf (pmf p) = p" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
251 |
by (intro measure_pmf_inject[THEN iffD1]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
252 |
(simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
253 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
254 |
fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
255 |
then show "pmf (embed_pmf f) = f" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
256 |
by (auto intro!: pmf_embed_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
257 |
qed (rule pmf_nonneg) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
258 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
259 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
260 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
261 |
locale pmf_as_function |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
262 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
263 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
264 |
setup_lifting td_pmf_embed_pmf |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
265 |
|
58730 | 266 |
lemma set_pmf_transfer[transfer_rule]: |
267 |
assumes "bi_total A" |
|
268 |
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf" |
|
269 |
using `bi_total A` |
|
270 |
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) |
|
271 |
metis+ |
|
272 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
273 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
274 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
275 |
(* |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
276 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
277 |
definition |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
278 |
"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)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
279 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
280 |
lift_definition pmf_join :: "real \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf" is |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
281 |
"\<lambda>p M1 M2. density (count_space UNIV) (\<lambda>x. p * measure M1 {x} + (1 - p) * measure M2 {x})" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
282 |
sorry |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
283 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
284 |
lift_definition pmf_single :: "'a \<Rightarrow> 'a pmf" is |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
285 |
"\<lambda>x. uniform_measure (count_space UNIV) {x}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
286 |
sorry |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
287 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
288 |
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
289 |
proof - |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
290 |
show "map_pmf id = id" by (rule map_pmf_id) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
291 |
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
292 |
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" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
293 |
by (intro map_pmg_cong refl) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
294 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
295 |
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
296 |
by (rule pmf_set_map) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
297 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
298 |
{ fix p :: "'s pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
299 |
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
300 |
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
301 |
(auto intro: countable_set_pmf inj_on_to_nat_on) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
302 |
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
303 |
by (metis Field_natLeq card_of_least natLeq_Well_order) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
304 |
finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . } |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
305 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
306 |
show "\<And>R. pmf_rel R = |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
307 |
(BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
308 |
BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
309 |
by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
310 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
311 |
{ let ?f = "map_pmf fst" and ?s = "map_pmf snd" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
312 |
fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and A assume "\<And>x y. (x, y) \<in> set_pmf A \<Longrightarrow> R x y" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
313 |
fix S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" and B assume "\<And>y z. (y, z) \<in> set_pmf B \<Longrightarrow> S y z" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
314 |
assume "?f B = ?s A" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
315 |
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" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
316 |
sorry } |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
317 |
oops |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
318 |
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)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
319 |
by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
320 |
qed (fact natLeq_card_order natLeq_cinfinite)+ |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
321 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
322 |
notepad |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
323 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
324 |
fix x y :: "nat \<Rightarrow> real" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
325 |
def IJz \<equiv> "rec_nat ((0, 0), \<lambda>_. 0) (\<lambda>n ((I, J), z). |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
326 |
let a = x I - (\<Sum>j<J. z (I, j)) ; b = y J - (\<Sum>i<I. z (i, J)) in |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
327 |
((if a \<le> b then I + 1 else I, if b \<le> a then J + 1 else J), z((I, J) := min a b)))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
328 |
def I == "fst \<circ> fst \<circ> IJz" def J == "snd \<circ> fst \<circ> IJz" def z == "snd \<circ> IJz" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
329 |
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))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
330 |
have IJz_0[simp]: "\<And>p. z 0 p = 0" "I 0 = 0" "J 0 = 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
331 |
by (simp_all add: I_def J_def z_def IJz_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
332 |
have z_Suc[simp]: "\<And>n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
333 |
by (simp add: z_def I_def J_def IJz_def Let_def split_beta) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
334 |
have I_Suc[simp]: "\<And>n. I (Suc n) = (if ?a n \<le> ?b n then I n + 1 else I n)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
335 |
by (simp add: z_def I_def J_def IJz_def Let_def split_beta) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
336 |
have J_Suc[simp]: "\<And>n. J (Suc n) = (if ?b n \<le> ?a n then J n + 1 else J n)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
337 |
by (simp add: z_def I_def J_def IJz_def Let_def split_beta) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
338 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
339 |
{ fix N have "\<And>p. z N p \<noteq> 0 \<Longrightarrow> \<exists>n<N. p = (I n, J n)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
340 |
by (induct N) (auto simp add: less_Suc_eq split: split_if_asm) } |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
341 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
342 |
{ fix i n assume "i < I n" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
343 |
then have "(\<Sum>j. z n (i, j)) = x i" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
344 |
oops |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
345 |
*) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
346 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
347 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
348 |