author | hoelzl |
Tue, 10 Mar 2015 11:56:32 +0100 | |
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parent 59664 | 224741ede5ae |
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permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Probability_Mass_Function.thy |
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Author: Johannes Hölzl, TU München |
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Author: Andreas Lochbihler, ETH Zurich |
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*) |
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section \<open> Probability mass function \<close> |
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theory Probability_Mass_Function |
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imports |
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Giry_Monad |
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"~~/src/HOL/Number_Theory/Binomial" |
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"~~/src/HOL/Library/Multiset" |
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begin |
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lemma AE_emeasure_singleton: |
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assumes x: "emeasure M {x} \<noteq> 0" and ae: "AE x in M. P x" shows "P x" |
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proof - |
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from x have x_M: "{x} \<in> sets M" |
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by (auto intro: emeasure_notin_sets) |
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from ae obtain N where N: "{x\<in>space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" |
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by (auto elim: AE_E) |
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{ assume "\<not> P x" |
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with x_M[THEN sets.sets_into_space] N have "emeasure M {x} \<le> emeasure M N" |
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by (intro emeasure_mono) auto |
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with x N have False |
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by (auto simp: emeasure_le_0_iff) } |
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then show "P x" by auto |
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qed |
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lemma AE_measure_singleton: "measure M {x} \<noteq> 0 \<Longrightarrow> AE x in M. P x \<Longrightarrow> P x" |
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by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty) |
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lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b" |
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using ereal_divide[of a b] by simp |
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lemma (in finite_measure) countable_support: |
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"countable {x. measure M {x} \<noteq> 0}" |
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proof cases |
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assume "measure M (space M) = 0" |
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with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}" |
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by auto |
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then show ?thesis |
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by simp |
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next |
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let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}" |
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assume "?M \<noteq> 0" |
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then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})" |
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using reals_Archimedean[of "?m x / ?M" for x] |
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by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff) |
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have **: "\<And>n. finite {x. ?M / Suc n < ?m x}" |
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proof (rule ccontr) |
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fix n assume "infinite {x. ?M / 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> ?M / Suc n \<le> ?m x" |
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by auto |
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{ fix x assume "x \<in> X" |
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from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff) |
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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 "?M < (\<Sum>x\<in>X. ?M / Suc n)" |
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using `?M \<noteq> 0` |
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by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg) |
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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 have "?M < measure M (\<Union>x\<in>X. {x})" . |
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moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M" |
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using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto |
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ultimately show False by simp |
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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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lemma (in finite_measure) AE_support_countable: |
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assumes [simp]: "sets M = UNIV" |
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shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))" |
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proof |
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assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)" |
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then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S" |
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by auto |
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then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) = |
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(\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)" |
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by (subst emeasure_UN_countable) |
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(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) |
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also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)" |
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by (auto intro!: nn_integral_cong split: split_indicator) |
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also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})" |
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by (subst emeasure_UN_countable) |
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(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) |
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also have "\<dots> = emeasure M (space M)" |
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using ae by (intro emeasure_eq_AE) auto |
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finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)" |
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by (simp add: emeasure_single_in_space cong: rev_conj_cong) |
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with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"] |
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have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0" |
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by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong) |
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then show "AE x in M. measure M {x} \<noteq> 0" |
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by (auto simp: emeasure_eq_measure) |
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qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support) |
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subsection \<open> PMF as measure \<close> |
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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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interpretation measure_pmf!: subprob_space "measure_pmf M" for M |
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by (rule prob_space_imp_subprob_space) unfold_locales |
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lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)" |
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by unfold_locales |
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locale pmf_as_measure |
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begin |
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setup_lifting type_definition_pmf |
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end |
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context |
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begin |
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interpretation pmf_as_measure . |
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lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" |
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by transfer blast |
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lemma sets_measure_pmf_count_space[measurable_cong]: |
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"sets (measure_pmf M) = sets (count_space UNIV)" |
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by simp |
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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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lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))" |
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by (simp add: space_subprob_algebra subprob_space_measure_pmf) |
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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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lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)" |
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by (intro measurable_cong_sets) simp_all |
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lemma measurable_pair_restrict_pmf2: |
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assumes "countable A" |
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assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L" |
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shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _") |
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proof - |
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have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
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by (simp add: restrict_count_space) |
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show ?thesis |
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by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A, |
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unfolded pair_collapse] assms) |
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measurable |
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qed |
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lemma measurable_pair_restrict_pmf1: |
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assumes "countable A" |
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assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L" |
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shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L" |
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proof - |
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have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
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by (simp add: restrict_count_space) |
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show ?thesis |
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by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, |
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unfolded pair_collapse] assms) |
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measurable |
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qed |
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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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declare [[coercion set_pmf]] |
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lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M" |
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by transfer simp |
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lemma emeasure_pmf_single_eq_zero_iff: |
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fixes M :: "'a pmf" |
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shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M" |
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by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
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lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)" |
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using AE_measure_singleton[of M] AE_measure_pmf[of M] |
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by (auto simp: set_pmf.rep_eq) |
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lemma countable_set_pmf [simp]: "countable (set_pmf p)" |
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by transfer (metis prob_space.finite_measure finite_measure.countable_support) |
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lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x" |
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by transfer (simp add: less_le measure_nonneg) |
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lemma pmf_nonneg: "0 \<le> pmf p x" |
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by transfer (simp add: measure_nonneg) |
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lemma pmf_le_1: "pmf p x \<le> 1" |
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by (simp add: pmf.rep_eq) |
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214 |
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}" |
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|
215 |
using AE_measure_pmf[of M] by (intro notI) simp |
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|
216 |
|
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|
217 |
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0" |
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|
218 |
by transfer simp |
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219 |
|
59664 | 220 |
lemma set_pmf_eq: "set_pmf M = {x. pmf M x \<noteq> 0}" |
221 |
by (auto simp: set_pmf_iff) |
|
222 |
||
223 |
lemma emeasure_pmf_single: |
|
224 |
fixes M :: "'a pmf" |
|
225 |
shows "emeasure M {x} = pmf M x" |
|
226 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
|
227 |
||
59000 | 228 |
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)" |
229 |
by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single) |
|
230 |
||
59023 | 231 |
lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S" |
59425 | 232 |
using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure) |
59023 | 233 |
|
59000 | 234 |
lemma nn_integral_measure_pmf_support: |
235 |
fixes f :: "'a \<Rightarrow> ereal" |
|
236 |
assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> set_pmf M \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0" |
|
237 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)" |
|
238 |
proof - |
|
239 |
have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
|
240 |
using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator) |
|
241 |
also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})" |
|
242 |
using assms by (intro nn_integral_indicator_finite) auto |
|
243 |
finally show ?thesis |
|
244 |
by (simp add: emeasure_measure_pmf_finite) |
|
245 |
qed |
|
246 |
||
247 |
lemma nn_integral_measure_pmf_finite: |
|
248 |
fixes f :: "'a \<Rightarrow> ereal" |
|
249 |
assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" |
|
250 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)" |
|
251 |
using assms by (intro nn_integral_measure_pmf_support) auto |
|
252 |
lemma integrable_measure_pmf_finite: |
|
253 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
|
254 |
shows "finite (set_pmf M) \<Longrightarrow> integrable M f" |
|
255 |
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite) |
|
256 |
||
257 |
lemma integral_measure_pmf: |
|
258 |
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A" |
|
259 |
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)" |
|
260 |
proof - |
|
261 |
have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)" |
|
262 |
using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff) |
|
263 |
also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)" |
|
264 |
by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite) |
|
265 |
finally show ?thesis . |
|
266 |
qed |
|
267 |
||
268 |
lemma integrable_pmf: "integrable (count_space X) (pmf M)" |
|
269 |
proof - |
|
270 |
have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))" |
|
271 |
by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator) |
|
272 |
then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)" |
|
273 |
by (simp add: integrable_iff_bounded pmf_nonneg) |
|
274 |
then show ?thesis |
|
59023 | 275 |
by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def) |
59000 | 276 |
qed |
277 |
||
278 |
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X" |
|
279 |
proof - |
|
280 |
have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)" |
|
281 |
by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral) |
|
282 |
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))" |
|
283 |
by (auto intro!: nn_integral_cong_AE split: split_indicator |
|
284 |
simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator |
|
285 |
AE_count_space set_pmf_iff) |
|
286 |
also have "\<dots> = emeasure M (X \<inter> M)" |
|
287 |
by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf) |
|
288 |
also have "\<dots> = emeasure M X" |
|
289 |
by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff) |
|
290 |
finally show ?thesis |
|
291 |
by (simp add: measure_pmf.emeasure_eq_measure) |
|
292 |
qed |
|
293 |
||
294 |
lemma integral_pmf_restrict: |
|
295 |
"(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow> |
|
296 |
(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)" |
|
297 |
by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff) |
|
298 |
||
58587
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299 |
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" |
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|
300 |
proof - |
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|
301 |
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" |
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|
302 |
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) |
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|
303 |
then show ?thesis |
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|
304 |
using measure_pmf.emeasure_space_1 by simp |
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|
305 |
qed |
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|
306 |
|
59490 | 307 |
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1" |
308 |
using measure_pmf.emeasure_space_1[of M] by simp |
|
309 |
||
59023 | 310 |
lemma in_null_sets_measure_pmfI: |
311 |
"A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)" |
|
312 |
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"] |
|
313 |
by(auto simp add: null_sets_def AE_measure_pmf_iff) |
|
314 |
||
59664 | 315 |
lemma measure_subprob: "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" |
316 |
by (simp add: space_subprob_algebra subprob_space_measure_pmf) |
|
317 |
||
318 |
subsection \<open> Monad Interpretation \<close> |
|
319 |
||
320 |
lemma measurable_measure_pmf[measurable]: |
|
321 |
"(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))" |
|
322 |
by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales |
|
323 |
||
324 |
lemma bind_measure_pmf_cong: |
|
325 |
assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)" |
|
326 |
assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" |
|
327 |
shows "bind (measure_pmf x) A = bind (measure_pmf x) B" |
|
328 |
proof (rule measure_eqI) |
|
329 |
show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)" |
|
330 |
using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra) |
|
331 |
next |
|
332 |
fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)" |
|
333 |
then have X: "X \<in> sets N" |
|
334 |
using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra) |
|
335 |
show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X" |
|
336 |
using assms |
|
337 |
by (subst (1 2) emeasure_bind[where N=N, OF _ _ X]) |
|
338 |
(auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
339 |
qed |
|
340 |
||
341 |
lift_definition bind_pmf :: "'a pmf \<Rightarrow> ('a \<Rightarrow> 'b pmf ) \<Rightarrow> 'b pmf" is bind |
|
342 |
proof (clarify, intro conjI) |
|
343 |
fix f :: "'a measure" and g :: "'a \<Rightarrow> 'b measure" |
|
344 |
assume "prob_space f" |
|
345 |
then interpret f: prob_space f . |
|
346 |
assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} \<noteq> 0" |
|
347 |
then have s_f[simp]: "sets f = sets (count_space UNIV)" |
|
348 |
by simp |
|
349 |
assume g: "\<And>x. prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. measure (g x) {y} \<noteq> 0)" |
|
350 |
then have g: "\<And>x. prob_space (g x)" and s_g[simp]: "\<And>x. sets (g x) = sets (count_space UNIV)" |
|
351 |
and ae_g: "\<And>x. AE y in g x. measure (g x) {y} \<noteq> 0" |
|
352 |
by auto |
|
353 |
||
354 |
have [measurable]: "g \<in> measurable f (subprob_algebra (count_space UNIV))" |
|
355 |
by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g) |
|
356 |
||
357 |
show "prob_space (f \<guillemotright>= g)" |
|
358 |
using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto |
|
359 |
then interpret fg: prob_space "f \<guillemotright>= g" . |
|
360 |
show [simp]: "sets (f \<guillemotright>= g) = UNIV" |
|
361 |
using sets_eq_imp_space_eq[OF s_f] |
|
362 |
by (subst sets_bind[where N="count_space UNIV"]) auto |
|
363 |
show "AE x in f \<guillemotright>= g. measure (f \<guillemotright>= g) {x} \<noteq> 0" |
|
364 |
apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"]) |
|
365 |
using ae_f |
|
366 |
apply eventually_elim |
|
367 |
using ae_g |
|
368 |
apply eventually_elim |
|
369 |
apply (auto dest: AE_measure_singleton) |
|
370 |
done |
|
371 |
qed |
|
372 |
||
373 |
lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)" |
|
374 |
unfolding pmf.rep_eq bind_pmf.rep_eq |
|
375 |
by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg |
|
376 |
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) |
|
377 |
||
378 |
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)" |
|
379 |
using ereal_pmf_bind[of N f i] |
|
380 |
by (subst (asm) nn_integral_eq_integral) |
|
381 |
(auto simp: pmf_nonneg pmf_le_1 |
|
382 |
intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) |
|
383 |
||
384 |
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c" |
|
385 |
by transfer (simp add: bind_const' prob_space_imp_subprob_space) |
|
386 |
||
59665 | 387 |
lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))" |
59664 | 388 |
unfolding set_pmf_eq ereal_eq_0(1)[symmetric] ereal_pmf_bind |
389 |
by (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff set_pmf_eq not_le less_le pmf_nonneg) |
|
390 |
||
391 |
lemma bind_pmf_cong: |
|
392 |
assumes "p = q" |
|
393 |
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" |
|
394 |
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq |
|
395 |
by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf |
|
396 |
sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"] |
|
397 |
intro!: nn_integral_cong_AE measure_eqI) |
|
398 |
||
399 |
lemma bind_pmf_cong_simp: |
|
400 |
"p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q =simp=> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g" |
|
401 |
by (simp add: simp_implies_def cong: bind_pmf_cong) |
|
402 |
||
403 |
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))" |
|
404 |
by transfer simp |
|
405 |
||
406 |
lemma nn_integral_bind_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>bind_pmf M N) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)" |
|
407 |
using measurable_measure_pmf[of N] |
|
408 |
unfolding measure_pmf_bind |
|
409 |
apply (subst (1 3) nn_integral_max_0[symmetric]) |
|
410 |
apply (intro nn_integral_bind[where B="count_space UNIV"]) |
|
411 |
apply auto |
|
412 |
done |
|
413 |
||
414 |
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)" |
|
415 |
using measurable_measure_pmf[of N] |
|
416 |
unfolding measure_pmf_bind |
|
417 |
by (subst emeasure_bind[where N="count_space UNIV"]) auto |
|
418 |
||
419 |
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)" |
|
420 |
by (auto intro!: prob_space_return simp: AE_return measure_return) |
|
421 |
||
422 |
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x" |
|
423 |
by transfer |
|
424 |
(auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"] |
|
425 |
simp: space_subprob_algebra) |
|
426 |
||
59665 | 427 |
lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}" |
59664 | 428 |
by transfer (auto simp add: measure_return split: split_indicator) |
429 |
||
430 |
lemma bind_return_pmf': "bind_pmf N return_pmf = N" |
|
431 |
proof (transfer, clarify) |
|
432 |
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N" |
|
433 |
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return') |
|
434 |
qed |
|
435 |
||
436 |
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)" |
|
437 |
by transfer |
|
438 |
(auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"] |
|
439 |
simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space) |
|
440 |
||
441 |
definition "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))" |
|
442 |
||
443 |
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))" |
|
444 |
by (simp add: map_pmf_def bind_assoc_pmf) |
|
445 |
||
446 |
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))" |
|
447 |
by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) |
|
448 |
||
449 |
lemma map_pmf_transfer[transfer_rule]: |
|
450 |
"rel_fun op = (rel_fun cr_pmf cr_pmf) (\<lambda>f M. distr M (count_space UNIV) f) map_pmf" |
|
451 |
proof - |
|
452 |
have "rel_fun op = (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf) |
|
453 |
(\<lambda>f M. M \<guillemotright>= (return (count_space UNIV) o f)) map_pmf" |
|
454 |
unfolding map_pmf_def[abs_def] comp_def by transfer_prover |
|
455 |
then show ?thesis |
|
456 |
by (force simp: rel_fun_def cr_pmf_def bind_return_distr) |
|
457 |
qed |
|
458 |
||
459 |
lemma map_pmf_rep_eq: |
|
460 |
"measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f" |
|
461 |
unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq |
|
462 |
using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def) |
|
463 |
||
58587
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
464 |
lemma map_pmf_id[simp]: "map_pmf id = id" |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
465 |
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
466 |
|
59053 | 467 |
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)" |
468 |
using map_pmf_id unfolding id_def . |
|
469 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
470 |
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
471 |
by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
472 |
|
59000 | 473 |
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M" |
474 |
using map_pmf_compose[of f g] by (simp add: comp_def) |
|
475 |
||
59664 | 476 |
lemma map_pmf_cong: "p = q \<Longrightarrow> (\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q" |
477 |
unfolding map_pmf_def by (rule bind_pmf_cong) auto |
|
478 |
||
479 |
lemma pmf_set_map: "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
|
59665 | 480 |
by (auto simp add: comp_def fun_eq_iff map_pmf_def) |
59664 | 481 |
|
59665 | 482 |
lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M" |
59664 | 483 |
using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
484 |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
485 |
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)" |
59664 | 486 |
unfolding map_pmf_rep_eq by (subst emeasure_distr) auto |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
487 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
488 |
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)" |
59664 | 489 |
unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
490 |
|
59023 | 491 |
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)" |
59664 | 492 |
proof (transfer fixing: f x) |
59023 | 493 |
fix p :: "'b measure" |
494 |
presume "prob_space p" |
|
495 |
then interpret prob_space p . |
|
496 |
presume "sets p = UNIV" |
|
497 |
then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))" |
|
498 |
by(simp add: measure_distr measurable_def emeasure_eq_measure) |
|
499 |
qed simp_all |
|
500 |
||
501 |
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A" |
|
502 |
proof - |
|
503 |
have "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = (\<integral>\<^sup>+ x. pmf p x \<partial>count_space (A \<inter> set_pmf p))" |
|
504 |
by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong) |
|
505 |
also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})" |
|
506 |
by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def) |
|
507 |
also have "\<dots> = emeasure (measure_pmf p) ((\<Union>x\<in>A \<inter> set_pmf p. {x}) \<union> {x. x \<in> A \<and> x \<notin> set_pmf p})" |
|
508 |
by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI) |
|
509 |
also have "\<dots> = emeasure (measure_pmf p) A" |
|
510 |
by(auto intro: arg_cong2[where f=emeasure]) |
|
511 |
finally show ?thesis . |
|
512 |
qed |
|
513 |
||
59664 | 514 |
lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)" |
515 |
by transfer (simp add: distr_return) |
|
516 |
||
517 |
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c" |
|
518 |
by transfer (auto simp: prob_space.distr_const) |
|
519 |
||
520 |
lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x" |
|
521 |
by transfer (simp add: measure_return) |
|
522 |
||
523 |
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x" |
|
524 |
unfolding return_pmf.rep_eq by (intro nn_integral_return) auto |
|
525 |
||
526 |
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x" |
|
527 |
unfolding return_pmf.rep_eq by (intro emeasure_return) auto |
|
528 |
||
529 |
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y" |
|
530 |
by (metis insertI1 set_return_pmf singletonD) |
|
531 |
||
59665 | 532 |
lemma map_pmf_eq_return_pmf_iff: |
533 |
"map_pmf f p = return_pmf x \<longleftrightarrow> (\<forall>y \<in> set_pmf p. f y = x)" |
|
534 |
proof |
|
535 |
assume "map_pmf f p = return_pmf x" |
|
536 |
then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp |
|
537 |
then show "\<forall>y \<in> set_pmf p. f y = x" by auto |
|
538 |
next |
|
539 |
assume "\<forall>y \<in> set_pmf p. f y = x" |
|
540 |
then show "map_pmf f p = return_pmf x" |
|
541 |
unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto |
|
542 |
qed |
|
543 |
||
59664 | 544 |
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))" |
545 |
||
546 |
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b" |
|
547 |
unfolding pair_pmf_def pmf_bind pmf_return |
|
548 |
apply (subst integral_measure_pmf[where A="{b}"]) |
|
549 |
apply (auto simp: indicator_eq_0_iff) |
|
550 |
apply (subst integral_measure_pmf[where A="{a}"]) |
|
551 |
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg) |
|
552 |
done |
|
553 |
||
59665 | 554 |
lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B" |
59664 | 555 |
unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto |
556 |
||
557 |
lemma measure_pmf_in_subprob_space[measurable (raw)]: |
|
558 |
"measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" |
|
559 |
by (simp add: space_subprob_algebra) intro_locales |
|
560 |
||
561 |
lemma nn_integral_pair_pmf': "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. f (a, b) \<partial>B \<partial>A)" |
|
562 |
proof - |
|
563 |
have "(\<integral>\<^sup>+x. f x \<partial>pair_pmf A B) = (\<integral>\<^sup>+x. max 0 (f x) * indicator (A \<times> B) x \<partial>pair_pmf A B)" |
|
564 |
by (subst nn_integral_max_0[symmetric]) |
|
59665 | 565 |
(auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE) |
59664 | 566 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)" |
567 |
by (simp add: pair_pmf_def) |
|
568 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)" |
|
569 |
by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
570 |
finally show ?thesis |
|
571 |
unfolding nn_integral_max_0 . |
|
572 |
qed |
|
573 |
||
574 |
lemma bind_pair_pmf: |
|
575 |
assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)" |
|
576 |
shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))" |
|
577 |
(is "?L = ?R") |
|
578 |
proof (rule measure_eqI) |
|
579 |
have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)" |
|
580 |
using M[THEN measurable_space] by (simp_all add: space_pair_measure) |
|
581 |
||
582 |
note measurable_bind[where N="count_space UNIV", measurable] |
|
583 |
note measure_pmf_in_subprob_space[simp] |
|
584 |
||
585 |
have sets_eq_N: "sets ?L = N" |
|
586 |
by (subst sets_bind[OF sets_kernel[OF M']]) auto |
|
587 |
show "sets ?L = sets ?R" |
|
588 |
using measurable_space[OF M] |
|
589 |
by (simp add: sets_eq_N space_pair_measure space_subprob_algebra) |
|
590 |
fix X assume "X \<in> sets ?L" |
|
591 |
then have X[measurable]: "X \<in> sets N" |
|
592 |
unfolding sets_eq_N . |
|
593 |
then show "emeasure ?L X = emeasure ?R X" |
|
594 |
apply (simp add: emeasure_bind[OF _ M' X]) |
|
595 |
apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A] |
|
59665 | 596 |
nn_integral_measure_pmf_finite emeasure_nonneg pmf_return one_ereal_def[symmetric]) |
59664 | 597 |
apply (subst emeasure_bind[OF _ _ X]) |
598 |
apply measurable |
|
599 |
apply (subst emeasure_bind[OF _ _ X]) |
|
600 |
apply measurable |
|
601 |
done |
|
602 |
qed |
|
603 |
||
604 |
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A" |
|
605 |
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
606 |
||
607 |
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B" |
|
608 |
by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
609 |
||
610 |
lemma nn_integral_pmf': |
|
611 |
"inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)" |
|
612 |
by (subst nn_integral_bij_count_space[where g=f and B="f`A"]) |
|
613 |
(auto simp: bij_betw_def nn_integral_pmf) |
|
614 |
||
615 |
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0" |
|
616 |
using pmf_nonneg[of M p] by simp |
|
617 |
||
618 |
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0" |
|
619 |
using pmf_nonneg[of M p] by simp_all |
|
620 |
||
621 |
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M" |
|
622 |
unfolding set_pmf_iff by simp |
|
623 |
||
624 |
lemma pmf_map_inj: "inj_on f (set_pmf M) \<Longrightarrow> x \<in> set_pmf M \<Longrightarrow> pmf (map_pmf f M) (f x) = pmf M x" |
|
625 |
by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD |
|
626 |
intro!: measure_pmf.finite_measure_eq_AE) |
|
627 |
||
628 |
subsection \<open> PMFs as function \<close> |
|
59000 | 629 |
|
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
630 |
context |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
631 |
fixes f :: "'a \<Rightarrow> real" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
632 |
assumes nonneg: "\<And>x. 0 \<le> f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
633 |
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
|
634 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
635 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
636 |
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
|
637 |
proof (intro conjI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
638 |
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
|
639 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
640 |
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
|
641 |
measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
59053
diff
changeset
|
642 |
by (simp add: AE_density nonneg measure_def emeasure_density max_def) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
643 |
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
|
644 |
by default (simp add: emeasure_density prob) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
645 |
qed simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
646 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
647 |
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
|
648 |
proof transfer |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
649 |
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
|
650 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
651 |
fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
59053
diff
changeset
|
652 |
by transfer (simp add: measure_def emeasure_density nonneg max_def) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
653 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
654 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
655 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
656 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
657 |
lemma embed_pmf_transfer: |
58730 | 658 |
"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
|
659 |
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
|
660 |
|
59000 | 661 |
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" |
662 |
proof (transfer, elim conjE) |
|
663 |
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
|
664 |
assume "prob_space M" then interpret prob_space M . |
|
665 |
show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))" |
|
666 |
proof (rule measure_eqI) |
|
667 |
fix A :: "'a set" |
|
668 |
have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = |
|
669 |
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)" |
|
670 |
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) |
|
671 |
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))" |
|
672 |
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) |
|
673 |
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})" |
|
674 |
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) |
|
675 |
(auto simp: disjoint_family_on_def) |
|
676 |
also have "\<dots> = emeasure M A" |
|
677 |
using ae by (intro emeasure_eq_AE) auto |
|
678 |
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A" |
|
679 |
using emeasure_space_1 by (simp add: emeasure_density) |
|
680 |
qed simp |
|
681 |
qed |
|
682 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
683 |
lemma td_pmf_embed_pmf: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
684 |
"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
|
685 |
unfolding type_definition_def |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
686 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
687 |
fix p :: "'a pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
688 |
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
|
689 |
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
|
690 |
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
|
691 |
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
|
692 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
693 |
show "embed_pmf (pmf p) = p" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
694 |
by (intro measure_pmf_inject[THEN iffD1]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
695 |
(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
|
696 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
697 |
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
|
698 |
then show "pmf (embed_pmf f) = f" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
699 |
by (auto intro!: pmf_embed_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
700 |
qed (rule pmf_nonneg) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
701 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
702 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
703 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
704 |
locale pmf_as_function |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
705 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
706 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
707 |
setup_lifting td_pmf_embed_pmf |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
708 |
|
58730 | 709 |
lemma set_pmf_transfer[transfer_rule]: |
710 |
assumes "bi_total A" |
|
711 |
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf" |
|
712 |
using `bi_total A` |
|
713 |
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) |
|
714 |
metis+ |
|
715 |
||
59000 | 716 |
end |
717 |
||
718 |
context |
|
719 |
begin |
|
720 |
||
721 |
interpretation pmf_as_function . |
|
722 |
||
723 |
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N" |
|
724 |
by transfer auto |
|
725 |
||
726 |
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)" |
|
727 |
by (auto intro: pmf_eqI) |
|
728 |
||
59664 | 729 |
lemma bind_commute_pmf: "bind_pmf A (\<lambda>x. bind_pmf B (C x)) = bind_pmf B (\<lambda>y. bind_pmf A (\<lambda>x. C x y))" |
730 |
unfolding pmf_eq_iff pmf_bind |
|
731 |
proof |
|
732 |
fix i |
|
733 |
interpret B: prob_space "restrict_space B B" |
|
734 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
735 |
(auto simp: AE_measure_pmf_iff) |
|
736 |
interpret A: prob_space "restrict_space A A" |
|
737 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
738 |
(auto simp: AE_measure_pmf_iff) |
|
739 |
||
740 |
interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B" |
|
741 |
by unfold_locales |
|
742 |
||
743 |
have "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>A)" |
|
744 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict) |
|
745 |
also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)" |
|
746 |
by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
|
747 |
countable_set_pmf borel_measurable_count_space) |
|
748 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)" |
|
749 |
by (rule AB.Fubini_integral[symmetric]) |
|
750 |
(auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2 |
|
751 |
simp: pmf_nonneg pmf_le_1 measurable_restrict_space1) |
|
752 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)" |
|
753 |
by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
|
754 |
countable_set_pmf borel_measurable_count_space) |
|
755 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" |
|
756 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric]) |
|
757 |
finally show "(\<integral> x. \<integral> y. pmf (C x y) i \<partial>B \<partial>A) = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" . |
|
758 |
qed |
|
759 |
||
760 |
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)" |
|
761 |
proof (safe intro!: pmf_eqI) |
|
762 |
fix a :: "'a" and b :: "'b" |
|
763 |
have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)" |
|
764 |
by (auto split: split_indicator) |
|
765 |
||
766 |
have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) = |
|
767 |
ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))" |
|
768 |
unfolding pmf_pair ereal_pmf_map |
|
769 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg |
|
770 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
771 |
then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)" |
|
772 |
by simp |
|
773 |
qed |
|
774 |
||
775 |
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)" |
|
776 |
proof (safe intro!: pmf_eqI) |
|
777 |
fix a :: "'a" and b :: "'b" |
|
778 |
have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)" |
|
779 |
by (auto split: split_indicator) |
|
780 |
||
781 |
have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) = |
|
782 |
ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))" |
|
783 |
unfolding pmf_pair ereal_pmf_map |
|
784 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg |
|
785 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
786 |
then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)" |
|
787 |
by simp |
|
788 |
qed |
|
789 |
||
790 |
lemma map_pair: "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)" |
|
791 |
by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta') |
|
792 |
||
59000 | 793 |
end |
794 |
||
59664 | 795 |
subsection \<open> Conditional Probabilities \<close> |
796 |
||
797 |
context |
|
798 |
fixes p :: "'a pmf" and s :: "'a set" |
|
799 |
assumes not_empty: "set_pmf p \<inter> s \<noteq> {}" |
|
800 |
begin |
|
801 |
||
802 |
interpretation pmf_as_measure . |
|
803 |
||
804 |
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0" |
|
805 |
proof |
|
806 |
assume "emeasure (measure_pmf p) s = 0" |
|
807 |
then have "AE x in measure_pmf p. x \<notin> s" |
|
808 |
by (rule AE_I[rotated]) auto |
|
809 |
with not_empty show False |
|
810 |
by (auto simp: AE_measure_pmf_iff) |
|
811 |
qed |
|
812 |
||
813 |
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0" |
|
814 |
using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp |
|
815 |
||
816 |
lift_definition cond_pmf :: "'a pmf" is |
|
817 |
"uniform_measure (measure_pmf p) s" |
|
818 |
proof (intro conjI) |
|
819 |
show "prob_space (uniform_measure (measure_pmf p) s)" |
|
820 |
by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero) |
|
821 |
show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0" |
|
822 |
by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure |
|
823 |
AE_measure_pmf_iff set_pmf.rep_eq) |
|
824 |
qed simp |
|
825 |
||
826 |
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)" |
|
827 |
by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq) |
|
828 |
||
59665 | 829 |
lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p \<inter> s" |
59664 | 830 |
by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm) |
831 |
||
832 |
end |
|
833 |
||
834 |
lemma cond_map_pmf: |
|
835 |
assumes "set_pmf p \<inter> f -` s \<noteq> {}" |
|
836 |
shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))" |
|
837 |
proof - |
|
838 |
have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}" |
|
59665 | 839 |
using assms by auto |
59664 | 840 |
{ fix x |
841 |
have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) = |
|
842 |
emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)" |
|
843 |
unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure) |
|
844 |
also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})" |
|
845 |
by auto |
|
846 |
also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) = |
|
847 |
ereal (pmf (cond_pmf (map_pmf f p) s) x)" |
|
848 |
using measure_measure_pmf_not_zero[OF *] |
|
849 |
by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric] |
|
850 |
del: ereal_divide) |
|
851 |
finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)" |
|
852 |
by simp } |
|
853 |
then show ?thesis |
|
854 |
by (intro pmf_eqI) simp |
|
855 |
qed |
|
856 |
||
857 |
lemma bind_cond_pmf_cancel: |
|
858 |
assumes in_S: "\<And>x. x \<in> set_pmf p \<Longrightarrow> x \<in> S x" "\<And>x. x \<in> set_pmf q \<Longrightarrow> x \<in> S x" |
|
859 |
assumes S_eq: "\<And>x y. x \<in> S y \<Longrightarrow> S x = S y" |
|
860 |
and same: "\<And>x. measure (measure_pmf p) (S x) = measure (measure_pmf q) (S x)" |
|
861 |
shows "bind_pmf p (\<lambda>x. cond_pmf q (S x)) = q" (is "?lhs = _") |
|
862 |
proof (rule pmf_eqI) |
|
863 |
{ fix x |
|
864 |
assume "x \<in> set_pmf p" |
|
865 |
hence "set_pmf p \<inter> (S x) \<noteq> {}" using in_S by auto |
|
866 |
hence "measure (measure_pmf p) (S x) \<noteq> 0" |
|
867 |
by(auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff) |
|
868 |
with same have "measure (measure_pmf q) (S x) \<noteq> 0" by simp |
|
869 |
hence "set_pmf q \<inter> S x \<noteq> {}" |
|
870 |
by(auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff) } |
|
871 |
note [simp] = this |
|
872 |
||
873 |
fix z |
|
874 |
have pmf_q_z: "z \<notin> S z \<Longrightarrow> pmf q z = 0" |
|
875 |
by(erule contrapos_np)(simp add: pmf_eq_0_set_pmf in_S) |
|
876 |
||
877 |
have "ereal (pmf ?lhs z) = \<integral>\<^sup>+ x. ereal (pmf (cond_pmf q (S x)) z) \<partial>measure_pmf p" |
|
878 |
by(simp add: ereal_pmf_bind) |
|
879 |
also have "\<dots> = \<integral>\<^sup>+ x. ereal (pmf q z / measure p (S z)) * indicator (S z) x \<partial>measure_pmf p" |
|
880 |
by(rule nn_integral_cong_AE)(auto simp add: AE_measure_pmf_iff pmf_cond same pmf_q_z in_S dest!: S_eq split: split_indicator) |
|
881 |
also have "\<dots> = pmf q z" using pmf_nonneg[of q z] |
|
882 |
by (subst nn_integral_cmult)(auto simp add: measure_nonneg measure_pmf.emeasure_eq_measure same measure_pmf.prob_eq_0 AE_measure_pmf_iff pmf_eq_0_set_pmf in_S) |
|
883 |
finally show "pmf ?lhs z = pmf q z" by simp |
|
884 |
qed |
|
885 |
||
886 |
subsection \<open> Relator \<close> |
|
887 |
||
888 |
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool" |
|
889 |
for R p q |
|
890 |
where |
|
891 |
"\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; |
|
892 |
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk> |
|
893 |
\<Longrightarrow> rel_pmf R p q" |
|
894 |
||
895 |
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf |
|
896 |
proof - |
|
897 |
show "map_pmf id = id" by (rule map_pmf_id) |
|
898 |
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) |
|
899 |
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" |
|
900 |
by (intro map_pmf_cong refl) |
|
901 |
||
902 |
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
|
903 |
by (rule pmf_set_map) |
|
904 |
||
905 |
{ fix p :: "'s pmf" |
|
906 |
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq" |
|
907 |
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) |
|
908 |
(auto intro: countable_set_pmf) |
|
909 |
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq" |
|
910 |
by (metis Field_natLeq card_of_least natLeq_Well_order) |
|
911 |
finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . } |
|
912 |
||
913 |
show "\<And>R. rel_pmf R = |
|
914 |
(BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO |
|
915 |
BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)" |
|
916 |
by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps) |
|
917 |
||
918 |
{ fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x |
|
919 |
assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z" |
|
920 |
and x: "x \<in> set_pmf p" |
|
921 |
thus "f x = g x" by simp } |
|
922 |
||
923 |
fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" |
|
924 |
{ fix p q r |
|
925 |
assume pq: "rel_pmf R p q" |
|
926 |
and qr:"rel_pmf S q r" |
|
927 |
from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
928 |
and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto |
|
929 |
from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z" |
|
930 |
and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto |
|
931 |
||
932 |
def pr \<equiv> "bind_pmf pq (\<lambda>(x, y). bind_pmf (cond_pmf qr {(y', z). y' = y}) (\<lambda>(y', z). return_pmf (x, z)))" |
|
933 |
have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {(y', z). y' = y} \<noteq> {}" |
|
59665 | 934 |
by (force simp: q') |
59664 | 935 |
|
936 |
have "rel_pmf (R OO S) p r" |
|
937 |
proof (rule rel_pmf.intros) |
|
938 |
fix x z assume "(x, z) \<in> pr" |
|
939 |
then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr" |
|
59665 | 940 |
by (auto simp: q pr_welldefined pr_def split_beta) |
59664 | 941 |
with pq qr show "(R OO S) x z" |
942 |
by blast |
|
943 |
next |
|
944 |
have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {(y', z). y' = y}))" |
|
945 |
by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_return_pmf) |
|
946 |
then show "map_pmf snd pr = r" |
|
947 |
unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) auto |
|
948 |
qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf map_pmf_def[symmetric] p) } |
|
949 |
then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)" |
|
950 |
by(auto simp add: le_fun_def) |
|
951 |
qed (fact natLeq_card_order natLeq_cinfinite)+ |
|
952 |
||
59665 | 953 |
lemma rel_pmf_conj[simp]: |
954 |
"rel_pmf (\<lambda>x y. P \<and> Q x y) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" |
|
955 |
"rel_pmf (\<lambda>x y. Q x y \<and> P) x y \<longleftrightarrow> P \<and> rel_pmf Q x y" |
|
956 |
using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+ |
|
957 |
||
958 |
lemma rel_pmf_top[simp]: "rel_pmf top = top" |
|
959 |
by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf |
|
960 |
intro: exI[of _ "pair_pmf x y" for x y]) |
|
961 |
||
59664 | 962 |
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)" |
963 |
proof safe |
|
964 |
fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M" |
|
965 |
then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b" |
|
966 |
and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq" |
|
967 |
by (force elim: rel_pmf.cases) |
|
968 |
moreover have "set_pmf (return_pmf x) = {x}" |
|
59665 | 969 |
by simp |
59664 | 970 |
with `a \<in> M` have "(x, a) \<in> pq" |
59665 | 971 |
by (force simp: eq) |
59664 | 972 |
with * show "R x a" |
973 |
by auto |
|
974 |
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"] |
|
59665 | 975 |
simp: map_fst_pair_pmf map_snd_pair_pmf) |
59664 | 976 |
|
977 |
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)" |
|
978 |
by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1) |
|
979 |
||
980 |
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2" |
|
981 |
unfolding rel_pmf_return_pmf2 set_return_pmf by simp |
|
982 |
||
983 |
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False" |
|
984 |
unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce |
|
985 |
||
986 |
lemma rel_pmf_rel_prod: |
|
987 |
"rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B') \<longleftrightarrow> rel_pmf R A B \<and> rel_pmf S A' B'" |
|
988 |
proof safe |
|
989 |
assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
990 |
then obtain pq where pq: "\<And>a b c d. ((a, c), (b, d)) \<in> set_pmf pq \<Longrightarrow> R a b \<and> S c d" |
|
991 |
and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'" |
|
992 |
by (force elim: rel_pmf.cases) |
|
993 |
show "rel_pmf R A B" |
|
994 |
proof (rule rel_pmf.intros) |
|
995 |
let ?f = "\<lambda>(a, b). (fst a, fst b)" |
|
996 |
have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd" |
|
997 |
by auto |
|
998 |
||
999 |
show "map_pmf fst (map_pmf ?f pq) = A" |
|
1000 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1001 |
show "map_pmf snd (map_pmf ?f pq) = B" |
|
1002 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1003 |
||
1004 |
fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)" |
|
1005 |
then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq" |
|
59665 | 1006 |
by auto |
59664 | 1007 |
from pq[OF this] show "R a b" .. |
1008 |
qed |
|
1009 |
show "rel_pmf S A' B'" |
|
1010 |
proof (rule rel_pmf.intros) |
|
1011 |
let ?f = "\<lambda>(a, b). (snd a, snd b)" |
|
1012 |
have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd" |
|
1013 |
by auto |
|
1014 |
||
1015 |
show "map_pmf fst (map_pmf ?f pq) = A'" |
|
1016 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1017 |
show "map_pmf snd (map_pmf ?f pq) = B'" |
|
1018 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1019 |
||
1020 |
fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)" |
|
1021 |
then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq" |
|
59665 | 1022 |
by auto |
59664 | 1023 |
from pq[OF this] show "S c d" .. |
1024 |
qed |
|
1025 |
next |
|
1026 |
assume "rel_pmf R A B" "rel_pmf S A' B'" |
|
1027 |
then obtain Rpq Spq |
|
1028 |
where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b" |
|
1029 |
"map_pmf fst Rpq = A" "map_pmf snd Rpq = B" |
|
1030 |
and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b" |
|
1031 |
"map_pmf fst Spq = A'" "map_pmf snd Spq = B'" |
|
1032 |
by (force elim: rel_pmf.cases) |
|
1033 |
||
1034 |
let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))" |
|
1035 |
let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)" |
|
1036 |
have [simp]: "(\<lambda>x. fst (?f x)) = (\<lambda>(a, b). (fst a, fst b))" "(\<lambda>x. snd (?f x)) = (\<lambda>(a, b). (snd a, snd b))" |
|
1037 |
by auto |
|
1038 |
||
1039 |
show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
1040 |
by (rule rel_pmf.intros[where pq="?pq"]) |
|
59665 | 1041 |
(auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq |
59664 | 1042 |
map_pair) |
1043 |
qed |
|
1044 |
||
1045 |
lemma rel_pmf_reflI: |
|
1046 |
assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" |
|
1047 |
shows "rel_pmf P p p" |
|
59665 | 1048 |
by (rule rel_pmf.intros[where pq="map_pmf (\<lambda>x. (x, x)) p"]) |
1049 |
(auto simp add: pmf.map_comp o_def assms) |
|
59664 | 1050 |
|
1051 |
context |
|
1052 |
begin |
|
1053 |
||
1054 |
interpretation pmf_as_measure . |
|
1055 |
||
1056 |
definition "join_pmf M = bind_pmf M (\<lambda>x. x)" |
|
1057 |
||
1058 |
lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)" |
|
1059 |
unfolding join_pmf_def bind_map_pmf .. |
|
1060 |
||
1061 |
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id" |
|
1062 |
by (simp add: join_pmf_def id_def) |
|
1063 |
||
1064 |
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)" |
|
1065 |
unfolding join_pmf_def pmf_bind .. |
|
1066 |
||
1067 |
lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)" |
|
1068 |
unfolding join_pmf_def ereal_pmf_bind .. |
|
1069 |
||
59665 | 1070 |
lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)" |
1071 |
by (simp add: join_pmf_def) |
|
59664 | 1072 |
|
1073 |
lemma join_return_pmf: "join_pmf (return_pmf M) = M" |
|
1074 |
by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq) |
|
1075 |
||
1076 |
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)" |
|
1077 |
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf) |
|
1078 |
||
1079 |
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A" |
|
1080 |
by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
1081 |
||
1082 |
end |
|
1083 |
||
1084 |
lemma rel_pmf_joinI: |
|
1085 |
assumes "rel_pmf (rel_pmf P) p q" |
|
1086 |
shows "rel_pmf P (join_pmf p) (join_pmf q)" |
|
1087 |
proof - |
|
1088 |
from assms obtain pq where p: "p = map_pmf fst pq" |
|
1089 |
and q: "q = map_pmf snd pq" |
|
1090 |
and P: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> rel_pmf P x y" |
|
1091 |
by cases auto |
|
1092 |
from P obtain PQ |
|
1093 |
where PQ: "\<And>x y a b. \<lbrakk> (x, y) \<in> set_pmf pq; (a, b) \<in> set_pmf (PQ x y) \<rbrakk> \<Longrightarrow> P a b" |
|
1094 |
and x: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf fst (PQ x y) = x" |
|
1095 |
and y: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> map_pmf snd (PQ x y) = y" |
|
1096 |
by(metis rel_pmf.simps) |
|
1097 |
||
1098 |
let ?r = "bind_pmf pq (\<lambda>(x, y). PQ x y)" |
|
59665 | 1099 |
have "\<And>a b. (a, b) \<in> set_pmf ?r \<Longrightarrow> P a b" by (auto intro: PQ) |
59664 | 1100 |
moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q" |
1101 |
by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong) |
|
1102 |
ultimately show ?thesis .. |
|
1103 |
qed |
|
1104 |
||
1105 |
lemma rel_pmf_bindI: |
|
1106 |
assumes pq: "rel_pmf R p q" |
|
1107 |
and fg: "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" |
|
1108 |
shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)" |
|
1109 |
unfolding bind_eq_join_pmf |
|
1110 |
by (rule rel_pmf_joinI) |
|
1111 |
(auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg) |
|
1112 |
||
1113 |
text {* |
|
1114 |
Proof that @{const rel_pmf} preserves orders. |
|
1115 |
Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism, |
|
1116 |
Theoretical Computer Science 12(1):19--37, 1980, |
|
1117 |
@{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"} |
|
1118 |
*} |
|
1119 |
||
1120 |
lemma |
|
1121 |
assumes *: "rel_pmf R p q" |
|
1122 |
and refl: "reflp R" and trans: "transp R" |
|
1123 |
shows measure_Ici: "measure p {y. R x y} \<le> measure q {y. R x y}" (is ?thesis1) |
|
1124 |
and measure_Ioi: "measure p {y. R x y \<and> \<not> R y x} \<le> measure q {y. R x y \<and> \<not> R y x}" (is ?thesis2) |
|
1125 |
proof - |
|
1126 |
from * obtain pq |
|
1127 |
where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
1128 |
and p: "p = map_pmf fst pq" |
|
1129 |
and q: "q = map_pmf snd pq" |
|
1130 |
by cases auto |
|
1131 |
show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans |
|
1132 |
by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE) |
|
1133 |
qed |
|
1134 |
||
1135 |
lemma rel_pmf_inf: |
|
1136 |
fixes p q :: "'a pmf" |
|
1137 |
assumes 1: "rel_pmf R p q" |
|
1138 |
assumes 2: "rel_pmf R q p" |
|
1139 |
and refl: "reflp R" and trans: "transp R" |
|
1140 |
shows "rel_pmf (inf R R\<inverse>\<inverse>) p q" |
|
1141 |
proof |
|
1142 |
let ?E = "\<lambda>x. {y. R x y \<and> R y x}" |
|
1143 |
let ?\<mu>E = "\<lambda>x. measure q (?E x)" |
|
1144 |
{ fix x |
|
1145 |
have "measure p (?E x) = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})" |
|
1146 |
by(auto intro!: arg_cong[where f="measure p"]) |
|
1147 |
also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}" |
|
1148 |
by (rule measure_pmf.finite_measure_Diff) auto |
|
1149 |
also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}" |
|
1150 |
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi) |
|
1151 |
also have "measure p {y. R x y} = measure q {y. R x y}" |
|
1152 |
using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici) |
|
1153 |
also have "measure q {y. R x y} - measure q {y. R x y \<and> ~ R y x} = |
|
1154 |
measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})" |
|
1155 |
by(rule measure_pmf.finite_measure_Diff[symmetric]) auto |
|
1156 |
also have "\<dots> = ?\<mu>E x" |
|
1157 |
by(auto intro!: arg_cong[where f="measure q"]) |
|
1158 |
also note calculation } |
|
1159 |
note eq = this |
|
1160 |
||
1161 |
def pq \<equiv> "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q (?E x)) (\<lambda>y. return_pmf (x, y)))" |
|
1162 |
||
1163 |
show "map_pmf fst pq = p" |
|
1164 |
by(simp add: pq_def map_bind_pmf map_return_pmf bind_return_pmf') |
|
1165 |
||
1166 |
show "map_pmf snd pq = q" |
|
1167 |
unfolding pq_def map_bind_pmf map_return_pmf bind_return_pmf' snd_conv |
|
1168 |
by(subst bind_cond_pmf_cancel)(auto simp add: reflpD[OF \<open>reflp R\<close>] eq intro: transpD[OF \<open>transp R\<close>]) |
|
1169 |
||
1170 |
fix x y |
|
1171 |
assume "(x, y) \<in> set_pmf pq" |
|
1172 |
moreover |
|
1173 |
{ assume "x \<in> set_pmf p" |
|
1174 |
hence "measure (measure_pmf p) (?E x) \<noteq> 0" |
|
59665 | 1175 |
by (auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff intro: reflpD[OF \<open>reflp R\<close>]) |
59664 | 1176 |
hence "measure (measure_pmf q) (?E x) \<noteq> 0" using eq by simp |
1177 |
hence "set_pmf q \<inter> {y. R x y \<and> R y x} \<noteq> {}" |
|
59665 | 1178 |
by (auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff) } |
59664 | 1179 |
ultimately show "inf R R\<inverse>\<inverse> x y" |
59665 | 1180 |
by (auto simp add: pq_def) |
59664 | 1181 |
qed |
1182 |
||
1183 |
lemma rel_pmf_antisym: |
|
1184 |
fixes p q :: "'a pmf" |
|
1185 |
assumes 1: "rel_pmf R p q" |
|
1186 |
assumes 2: "rel_pmf R q p" |
|
1187 |
and refl: "reflp R" and trans: "transp R" and antisym: "antisymP R" |
|
1188 |
shows "p = q" |
|
1189 |
proof - |
|
1190 |
from 1 2 refl trans have "rel_pmf (inf R R\<inverse>\<inverse>) p q" by(rule rel_pmf_inf) |
|
1191 |
also have "inf R R\<inverse>\<inverse> = op =" |
|
59665 | 1192 |
using refl antisym by (auto intro!: ext simp add: reflpD dest: antisymD) |
59664 | 1193 |
finally show ?thesis unfolding pmf.rel_eq . |
1194 |
qed |
|
1195 |
||
1196 |
lemma reflp_rel_pmf: "reflp R \<Longrightarrow> reflp (rel_pmf R)" |
|
1197 |
by(blast intro: reflpI rel_pmf_reflI reflpD) |
|
1198 |
||
1199 |
lemma antisymP_rel_pmf: |
|
1200 |
"\<lbrakk> reflp R; transp R; antisymP R \<rbrakk> |
|
1201 |
\<Longrightarrow> antisymP (rel_pmf R)" |
|
1202 |
by(rule antisymI)(blast intro: rel_pmf_antisym) |
|
1203 |
||
1204 |
lemma transp_rel_pmf: |
|
1205 |
assumes "transp R" |
|
1206 |
shows "transp (rel_pmf R)" |
|
1207 |
proof (rule transpI) |
|
1208 |
fix x y z |
|
1209 |
assume "rel_pmf R x y" and "rel_pmf R y z" |
|
1210 |
hence "rel_pmf (R OO R) x z" by (simp add: pmf.rel_compp relcompp.relcompI) |
|
1211 |
thus "rel_pmf R x z" |
|
1212 |
using assms by (metis (no_types) pmf.rel_mono rev_predicate2D transp_relcompp_less_eq) |
|
1213 |
qed |
|
1214 |
||
1215 |
subsection \<open> Distributions \<close> |
|
1216 |
||
59000 | 1217 |
context |
1218 |
begin |
|
1219 |
||
1220 |
interpretation pmf_as_function . |
|
1221 |
||
59093 | 1222 |
subsubsection \<open> Bernoulli Distribution \<close> |
1223 |
||
59000 | 1224 |
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is |
1225 |
"\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p" |
|
1226 |
by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool |
|
1227 |
split: split_max split_min) |
|
1228 |
||
1229 |
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p" |
|
1230 |
by transfer simp |
|
1231 |
||
1232 |
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p" |
|
1233 |
by transfer simp |
|
1234 |
||
1235 |
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV" |
|
1236 |
by (auto simp add: set_pmf_iff UNIV_bool) |
|
1237 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1238 |
lemma nn_integral_bernoulli_pmf[simp]: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1239 |
assumes [simp]: "0 \<le> p" "p \<le> 1" "\<And>x. 0 \<le> f x" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1240 |
shows "(\<integral>\<^sup>+x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1241 |
by (subst nn_integral_measure_pmf_support[of UNIV]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1242 |
(auto simp: UNIV_bool field_simps) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1243 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1244 |
lemma integral_bernoulli_pmf[simp]: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
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diff
changeset
|
1245 |
assumes [simp]: "0 \<le> p" "p \<le> 1" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1246 |
shows "(\<integral>x. f x \<partial>bernoulli_pmf p) = f True * p + f False * (1 - p)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1247 |
by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1248 |
|
59525 | 1249 |
lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2" |
1250 |
by(cases x) simp_all |
|
1251 |
||
1252 |
lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV" |
|
1253 |
by(rule measure_eqI)(simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure nn_integral_count_space_finite sets_uniform_count_measure) |
|
1254 |
||
59093 | 1255 |
subsubsection \<open> Geometric Distribution \<close> |
1256 |
||
59000 | 1257 |
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n" |
1258 |
proof |
|
1259 |
note geometric_sums[of "1 / 2"] |
|
1260 |
note sums_mult[OF this, of "1 / 2"] |
|
1261 |
from sums_suminf_ereal[OF this] |
|
1262 |
show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1" |
|
1263 |
by (simp add: nn_integral_count_space_nat field_simps) |
|
1264 |
qed simp |
|
1265 |
||
1266 |
lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n" |
|
1267 |
by transfer rule |
|
1268 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1269 |
lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV" |
59000 | 1270 |
by (auto simp: set_pmf_iff) |
1271 |
||
59093 | 1272 |
subsubsection \<open> Uniform Multiset Distribution \<close> |
1273 |
||
59000 | 1274 |
context |
1275 |
fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}" |
|
1276 |
begin |
|
1277 |
||
1278 |
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M" |
|
1279 |
proof |
|
1280 |
show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1" |
|
1281 |
using M_not_empty |
|
1282 |
by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size |
|
1283 |
setsum_divide_distrib[symmetric]) |
|
1284 |
(auto simp: size_multiset_overloaded_eq intro!: setsum.cong) |
|
1285 |
qed simp |
|
1286 |
||
1287 |
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M" |
|
1288 |
by transfer rule |
|
1289 |
||
1290 |
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M" |
|
1291 |
by (auto simp: set_pmf_iff) |
|
1292 |
||
1293 |
end |
|
1294 |
||
59093 | 1295 |
subsubsection \<open> Uniform Distribution \<close> |
1296 |
||
59000 | 1297 |
context |
1298 |
fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S" |
|
1299 |
begin |
|
1300 |
||
1301 |
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S" |
|
1302 |
proof |
|
1303 |
show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1" |
|
1304 |
using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto |
|
1305 |
qed simp |
|
1306 |
||
1307 |
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S" |
|
1308 |
by transfer rule |
|
1309 |
||
1310 |
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S" |
|
1311 |
using S_finite S_not_empty by (auto simp: set_pmf_iff) |
|
1312 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1313 |
lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1314 |
by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1315 |
|
59000 | 1316 |
end |
1317 |
||
59093 | 1318 |
subsubsection \<open> Poisson Distribution \<close> |
1319 |
||
1320 |
context |
|
1321 |
fixes rate :: real assumes rate_pos: "0 < rate" |
|
1322 |
begin |
|
1323 |
||
1324 |
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)" |
|
1325 |
proof |
|
1326 |
(* Proof by Manuel Eberl *) |
|
1327 |
||
1328 |
have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp |
|
59557 | 1329 |
by (simp add: field_simps divide_inverse [symmetric]) |
59093 | 1330 |
have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) = |
1331 |
exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)" |
|
1332 |
by (simp add: field_simps nn_integral_cmult[symmetric]) |
|
1333 |
also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)" |
|
1334 |
by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite) |
|
1335 |
also have "... = exp rate" unfolding exp_def |
|
59557 | 1336 |
by (simp add: field_simps divide_inverse [symmetric] transfer_int_nat_factorial) |
59093 | 1337 |
also have "ereal (exp (-rate)) * ereal (exp rate) = 1" |
1338 |
by (simp add: mult_exp_exp) |
|
1339 |
finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" . |
|
1340 |
qed (simp add: rate_pos[THEN less_imp_le]) |
|
1341 |
||
1342 |
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)" |
|
1343 |
by transfer rule |
|
1344 |
||
1345 |
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV" |
|
1346 |
using rate_pos by (auto simp: set_pmf_iff) |
|
1347 |
||
59000 | 1348 |
end |
1349 |
||
59093 | 1350 |
subsubsection \<open> Binomial Distribution \<close> |
1351 |
||
1352 |
context |
|
1353 |
fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1" |
|
1354 |
begin |
|
1355 |
||
1356 |
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)" |
|
1357 |
proof |
|
1358 |
have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) = |
|
1359 |
ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))" |
|
1360 |
using p_le_1 p_nonneg by (subst nn_integral_count_space') auto |
|
1361 |
also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n" |
|
1362 |
by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def) |
|
1363 |
finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1" |
|
1364 |
by simp |
|
1365 |
qed (insert p_nonneg p_le_1, simp) |
|
1366 |
||
1367 |
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)" |
|
1368 |
by transfer rule |
|
1369 |
||
1370 |
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})" |
|
1371 |
using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff) |
|
1372 |
||
1373 |
end |
|
1374 |
||
1375 |
end |
|
1376 |
||
1377 |
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}" |
|
1378 |
by (simp add: set_pmf_binomial_eq) |
|
1379 |
||
1380 |
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}" |
|
1381 |
by (simp add: set_pmf_binomial_eq) |
|
1382 |
||
1383 |
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}" |
|
1384 |
by (simp add: set_pmf_binomial_eq) |
|
1385 |
||
59000 | 1386 |
end |