author | hoelzl |
Tue, 10 Feb 2015 14:06:57 +0100 | |
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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 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 {* PMF as measure *} |
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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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lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" . |
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lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" . |
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lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is |
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"\<lambda>f M. distr M (count_space UNIV) f" |
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proof safe |
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fix M and f :: "'a \<Rightarrow> 'b" |
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let ?D = "distr M (count_space UNIV) f" |
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assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
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interpret prob_space M by fact |
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from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0" |
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proof eventually_elim |
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fix x |
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have "measure M {x} \<le> measure M (f -` {f x})" |
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by (intro finite_measure_mono) auto |
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then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0" |
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using measure_nonneg[of M "{x}"] by auto |
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qed |
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then show "AE x in ?D. measure ?D {x} \<noteq> 0" |
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by (simp add: AE_distr_iff measure_distr measurable_def) |
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qed (auto simp: measurable_def prob_space.prob_space_distr) |
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declare [[coercion set_pmf]] |
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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 sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" |
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by transfer metis |
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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 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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lemma emeasure_pmf_single: |
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177 |
fixes M :: "'a pmf" |
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hoelzl
parents:
diff
changeset
|
178 |
shows "emeasure M {x} = pmf M x" |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
179 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
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hoelzl
parents:
diff
changeset
|
180 |
|
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hoelzl
parents:
diff
changeset
|
181 |
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M" |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
182 |
by transfer simp |
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hoelzl
parents:
diff
changeset
|
183 |
|
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
184 |
lemma emeasure_pmf_single_eq_zero_iff: |
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hoelzl
parents:
diff
changeset
|
185 |
fixes M :: "'a pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
186 |
shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
187 |
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
188 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
189 |
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)" |
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add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
190 |
proof - |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
191 |
{ fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y" |
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hoelzl
parents:
diff
changeset
|
192 |
with P have "AE x in M. x \<noteq> y" |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
193 |
by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
194 |
with y have False |
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hoelzl
parents:
diff
changeset
|
195 |
by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) } |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
196 |
then show ?thesis |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
197 |
using AE_measure_pmf[of M] by auto |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
198 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
199 |
|
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
200 |
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}" |
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hoelzl
parents:
diff
changeset
|
201 |
using AE_measure_pmf[of M] by (intro notI) simp |
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
202 |
|
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
203 |
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0" |
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hoelzl
parents:
diff
changeset
|
204 |
by transfer simp |
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hoelzl
parents:
diff
changeset
|
205 |
|
59000 | 206 |
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)" |
207 |
by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single) |
|
208 |
||
59023 | 209 |
lemma measure_measure_pmf_finite: "finite S \<Longrightarrow> measure (measure_pmf M) S = setsum (pmf M) S" |
59425 | 210 |
using emeasure_measure_pmf_finite[of S M] by(simp add: measure_pmf.emeasure_eq_measure) |
59023 | 211 |
|
59000 | 212 |
lemma nn_integral_measure_pmf_support: |
213 |
fixes f :: "'a \<Rightarrow> ereal" |
|
214 |
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" |
|
215 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)" |
|
216 |
proof - |
|
217 |
have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
|
218 |
using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator) |
|
219 |
also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})" |
|
220 |
using assms by (intro nn_integral_indicator_finite) auto |
|
221 |
finally show ?thesis |
|
222 |
by (simp add: emeasure_measure_pmf_finite) |
|
223 |
qed |
|
224 |
||
225 |
lemma nn_integral_measure_pmf_finite: |
|
226 |
fixes f :: "'a \<Rightarrow> ereal" |
|
227 |
assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" |
|
228 |
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)" |
|
229 |
using assms by (intro nn_integral_measure_pmf_support) auto |
|
230 |
lemma integrable_measure_pmf_finite: |
|
231 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
|
232 |
shows "finite (set_pmf M) \<Longrightarrow> integrable M f" |
|
233 |
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite) |
|
234 |
||
235 |
lemma integral_measure_pmf: |
|
236 |
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A" |
|
237 |
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)" |
|
238 |
proof - |
|
239 |
have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)" |
|
240 |
using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff) |
|
241 |
also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)" |
|
242 |
by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite) |
|
243 |
finally show ?thesis . |
|
244 |
qed |
|
245 |
||
246 |
lemma integrable_pmf: "integrable (count_space X) (pmf M)" |
|
247 |
proof - |
|
248 |
have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))" |
|
249 |
by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator) |
|
250 |
then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)" |
|
251 |
by (simp add: integrable_iff_bounded pmf_nonneg) |
|
252 |
then show ?thesis |
|
59023 | 253 |
by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def) |
59000 | 254 |
qed |
255 |
||
256 |
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X" |
|
257 |
proof - |
|
258 |
have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)" |
|
259 |
by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral) |
|
260 |
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))" |
|
261 |
by (auto intro!: nn_integral_cong_AE split: split_indicator |
|
262 |
simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator |
|
263 |
AE_count_space set_pmf_iff) |
|
264 |
also have "\<dots> = emeasure M (X \<inter> M)" |
|
265 |
by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf) |
|
266 |
also have "\<dots> = emeasure M X" |
|
267 |
by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff) |
|
268 |
finally show ?thesis |
|
269 |
by (simp add: measure_pmf.emeasure_eq_measure) |
|
270 |
qed |
|
271 |
||
272 |
lemma integral_pmf_restrict: |
|
273 |
"(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow> |
|
274 |
(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)" |
|
275 |
by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff) |
|
276 |
||
58587
5484f6079bcd
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hoelzl
parents:
diff
changeset
|
277 |
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
278 |
proof - |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
279 |
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
280 |
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
281 |
then show ?thesis |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
282 |
using measure_pmf.emeasure_space_1 by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
283 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
284 |
|
59490 | 285 |
lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1" |
286 |
using measure_pmf.emeasure_space_1[of M] by simp |
|
287 |
||
59023 | 288 |
lemma in_null_sets_measure_pmfI: |
289 |
"A \<inter> set_pmf p = {} \<Longrightarrow> A \<in> null_sets (measure_pmf p)" |
|
290 |
using emeasure_eq_0_AE[where ?P="\<lambda>x. x \<in> A" and M="measure_pmf p"] |
|
291 |
by(auto simp add: null_sets_def AE_measure_pmf_iff) |
|
292 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
293 |
lemma map_pmf_id[simp]: "map_pmf id = id" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
294 |
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
295 |
|
59053 | 296 |
lemma map_pmf_ident[simp]: "map_pmf (\<lambda>x. x) = (\<lambda>x. x)" |
297 |
using map_pmf_id unfolding id_def . |
|
298 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
299 |
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
300 |
by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
301 |
|
59000 | 302 |
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M" |
303 |
using map_pmf_compose[of f g] by (simp add: comp_def) |
|
304 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
305 |
lemma map_pmf_cong: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
306 |
assumes "p = q" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
307 |
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
308 |
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
309 |
by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
310 |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
311 |
lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
312 |
unfolding map_pmf.rep_eq by (subst emeasure_distr) auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
313 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
314 |
lemma nn_integral_map_pmf[simp]: "(\<integral>\<^sup>+x. f x \<partial>map_pmf g M) = (\<integral>\<^sup>+x. f (g x) \<partial>M)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
315 |
unfolding map_pmf.rep_eq by (intro nn_integral_distr) auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
316 |
|
59023 | 317 |
lemma ereal_pmf_map: "pmf (map_pmf f p) x = (\<integral>\<^sup>+ y. indicator (f -` {x}) y \<partial>measure_pmf p)" |
318 |
proof(transfer fixing: f x) |
|
319 |
fix p :: "'b measure" |
|
320 |
presume "prob_space p" |
|
321 |
then interpret prob_space p . |
|
322 |
presume "sets p = UNIV" |
|
323 |
then show "ereal (measure (distr p (count_space UNIV) f) {x}) = integral\<^sup>N p (indicator (f -` {x}))" |
|
324 |
by(simp add: measure_distr measurable_def emeasure_eq_measure) |
|
325 |
qed simp_all |
|
326 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
327 |
lemma pmf_set_map: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
328 |
fixes f :: "'a \<Rightarrow> 'b" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
329 |
shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
330 |
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
331 |
fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
332 |
assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
333 |
interpret prob_space M by fact |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
334 |
show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
335 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
336 |
fix x assume "measure M (f -` {x}) \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
337 |
moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
338 |
using ae by (intro finite_measure_eq_AE) auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
339 |
ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
340 |
by (metis measure_empty) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
341 |
then show "x \<in> f ` {x. measure M {x} \<noteq> 0}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
342 |
by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
343 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
344 |
fix x assume "measure M {x} \<noteq> 0" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
345 |
then have "0 < measure M {x}" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
346 |
using measure_nonneg[of M "{x}"] by auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
347 |
also have "measure M {x} \<le> measure M (f -` {f x})" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
348 |
by (intro finite_measure_mono) auto |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
349 |
finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
350 |
by simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
351 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
352 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
353 |
|
59000 | 354 |
lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M" |
355 |
using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff) |
|
356 |
||
59023 | 357 |
lemma nn_integral_pmf: "(\<integral>\<^sup>+ x. pmf p x \<partial>count_space A) = emeasure (measure_pmf p) A" |
358 |
proof - |
|
359 |
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))" |
|
360 |
by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong) |
|
361 |
also have "\<dots> = emeasure (measure_pmf p) (\<Union>x\<in>A \<inter> set_pmf p. {x})" |
|
362 |
by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def) |
|
363 |
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})" |
|
364 |
by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI) |
|
365 |
also have "\<dots> = emeasure (measure_pmf p) A" |
|
366 |
by(auto intro: arg_cong2[where f=emeasure]) |
|
367 |
finally show ?thesis . |
|
368 |
qed |
|
369 |
||
59000 | 370 |
subsection {* PMFs as function *} |
371 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
372 |
context |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
373 |
fixes f :: "'a \<Rightarrow> real" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
374 |
assumes nonneg: "\<And>x. 0 \<le> f x" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
375 |
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
|
376 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
377 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
378 |
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
|
379 |
proof (intro conjI) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
380 |
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
|
381 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
382 |
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
|
383 |
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
|
384 |
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
|
385 |
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
|
386 |
by default (simp add: emeasure_density prob) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
387 |
qed simp |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
388 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
389 |
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
|
390 |
proof transfer |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
391 |
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
|
392 |
by (simp split: split_indicator) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
393 |
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
|
394 |
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
|
395 |
qed |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
396 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
397 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
398 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
399 |
lemma embed_pmf_transfer: |
58730 | 400 |
"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
|
401 |
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
|
402 |
|
59000 | 403 |
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" |
404 |
proof (transfer, elim conjE) |
|
405 |
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0" |
|
406 |
assume "prob_space M" then interpret prob_space M . |
|
407 |
show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))" |
|
408 |
proof (rule measure_eqI) |
|
409 |
fix A :: "'a set" |
|
410 |
have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = |
|
411 |
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)" |
|
412 |
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) |
|
413 |
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))" |
|
414 |
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) |
|
415 |
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})" |
|
416 |
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) |
|
417 |
(auto simp: disjoint_family_on_def) |
|
418 |
also have "\<dots> = emeasure M A" |
|
419 |
using ae by (intro emeasure_eq_AE) auto |
|
420 |
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A" |
|
421 |
using emeasure_space_1 by (simp add: emeasure_density) |
|
422 |
qed simp |
|
423 |
qed |
|
424 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
425 |
lemma td_pmf_embed_pmf: |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
426 |
"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
|
427 |
unfolding type_definition_def |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
428 |
proof safe |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
429 |
fix p :: "'a pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
430 |
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
|
431 |
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
|
432 |
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
|
433 |
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
|
434 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
435 |
show "embed_pmf (pmf p) = p" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
436 |
by (intro measure_pmf_inject[THEN iffD1]) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
437 |
(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
|
438 |
next |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
439 |
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
|
440 |
then show "pmf (embed_pmf f) = f" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
441 |
by (auto intro!: pmf_embed_pmf) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
442 |
qed (rule pmf_nonneg) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
443 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
444 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
445 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
446 |
locale pmf_as_function |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
447 |
begin |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
448 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
449 |
setup_lifting td_pmf_embed_pmf |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
450 |
|
58730 | 451 |
lemma set_pmf_transfer[transfer_rule]: |
452 |
assumes "bi_total A" |
|
453 |
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf" |
|
454 |
using `bi_total A` |
|
455 |
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) |
|
456 |
metis+ |
|
457 |
||
59000 | 458 |
end |
459 |
||
460 |
context |
|
461 |
begin |
|
462 |
||
463 |
interpretation pmf_as_function . |
|
464 |
||
465 |
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N" |
|
466 |
by transfer auto |
|
467 |
||
468 |
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)" |
|
469 |
by (auto intro: pmf_eqI) |
|
470 |
||
471 |
end |
|
472 |
||
473 |
context |
|
474 |
begin |
|
475 |
||
476 |
interpretation pmf_as_function . |
|
477 |
||
59093 | 478 |
subsubsection \<open> Bernoulli Distribution \<close> |
479 |
||
59000 | 480 |
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is |
481 |
"\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p" |
|
482 |
by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool |
|
483 |
split: split_max split_min) |
|
484 |
||
485 |
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p" |
|
486 |
by transfer simp |
|
487 |
||
488 |
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p" |
|
489 |
by transfer simp |
|
490 |
||
491 |
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV" |
|
492 |
by (auto simp add: set_pmf_iff UNIV_bool) |
|
493 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
494 |
lemma nn_integral_bernoulli_pmf[simp]: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
495 |
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
|
496 |
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
|
497 |
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
|
498 |
(auto simp: UNIV_bool field_simps) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
499 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
500 |
lemma integral_bernoulli_pmf[simp]: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
501 |
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
|
502 |
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
|
503 |
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
|
504 |
|
59093 | 505 |
subsubsection \<open> Geometric Distribution \<close> |
506 |
||
59000 | 507 |
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n" |
508 |
proof |
|
509 |
note geometric_sums[of "1 / 2"] |
|
510 |
note sums_mult[OF this, of "1 / 2"] |
|
511 |
from sums_suminf_ereal[OF this] |
|
512 |
show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1" |
|
513 |
by (simp add: nn_integral_count_space_nat field_simps) |
|
514 |
qed simp |
|
515 |
||
516 |
lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n" |
|
517 |
by transfer rule |
|
518 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
519 |
lemma set_pmf_geometric[simp]: "set_pmf geometric_pmf = UNIV" |
59000 | 520 |
by (auto simp: set_pmf_iff) |
521 |
||
59093 | 522 |
subsubsection \<open> Uniform Multiset Distribution \<close> |
523 |
||
59000 | 524 |
context |
525 |
fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}" |
|
526 |
begin |
|
527 |
||
528 |
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M" |
|
529 |
proof |
|
530 |
show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1" |
|
531 |
using M_not_empty |
|
532 |
by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size |
|
533 |
setsum_divide_distrib[symmetric]) |
|
534 |
(auto simp: size_multiset_overloaded_eq intro!: setsum.cong) |
|
535 |
qed simp |
|
536 |
||
537 |
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M" |
|
538 |
by transfer rule |
|
539 |
||
540 |
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M" |
|
541 |
by (auto simp: set_pmf_iff) |
|
542 |
||
543 |
end |
|
544 |
||
59093 | 545 |
subsubsection \<open> Uniform Distribution \<close> |
546 |
||
59000 | 547 |
context |
548 |
fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S" |
|
549 |
begin |
|
550 |
||
551 |
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S" |
|
552 |
proof |
|
553 |
show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1" |
|
554 |
using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto |
|
555 |
qed simp |
|
556 |
||
557 |
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S" |
|
558 |
by transfer rule |
|
559 |
||
560 |
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S" |
|
561 |
using S_finite S_not_empty by (auto simp: set_pmf_iff) |
|
562 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
563 |
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
|
564 |
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
|
565 |
|
59000 | 566 |
end |
567 |
||
59093 | 568 |
subsubsection \<open> Poisson Distribution \<close> |
569 |
||
570 |
context |
|
571 |
fixes rate :: real assumes rate_pos: "0 < rate" |
|
572 |
begin |
|
573 |
||
574 |
lift_definition poisson_pmf :: "nat pmf" is "\<lambda>k. rate ^ k / fact k * exp (-rate)" |
|
575 |
proof |
|
576 |
(* Proof by Manuel Eberl *) |
|
577 |
||
578 |
have summable: "summable (\<lambda>x::nat. rate ^ x / fact x)" using summable_exp |
|
579 |
by (simp add: field_simps field_divide_inverse[symmetric]) |
|
580 |
have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x * exp (-rate) \<partial>count_space UNIV) = |
|
581 |
exp (-rate) * (\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV)" |
|
582 |
by (simp add: field_simps nn_integral_cmult[symmetric]) |
|
583 |
also from rate_pos have "(\<integral>\<^sup>+(x::nat). rate ^ x / fact x \<partial>count_space UNIV) = (\<Sum>x. rate ^ x / fact x)" |
|
584 |
by (simp_all add: nn_integral_count_space_nat suminf_ereal summable suminf_ereal_finite) |
|
585 |
also have "... = exp rate" unfolding exp_def |
|
586 |
by (simp add: field_simps field_divide_inverse[symmetric] transfer_int_nat_factorial) |
|
587 |
also have "ereal (exp (-rate)) * ereal (exp rate) = 1" |
|
588 |
by (simp add: mult_exp_exp) |
|
589 |
finally show "(\<integral>\<^sup>+ x. ereal (rate ^ x / real (fact x) * exp (- rate)) \<partial>count_space UNIV) = 1" . |
|
590 |
qed (simp add: rate_pos[THEN less_imp_le]) |
|
591 |
||
592 |
lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)" |
|
593 |
by transfer rule |
|
594 |
||
595 |
lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV" |
|
596 |
using rate_pos by (auto simp: set_pmf_iff) |
|
597 |
||
59000 | 598 |
end |
599 |
||
59093 | 600 |
subsubsection \<open> Binomial Distribution \<close> |
601 |
||
602 |
context |
|
603 |
fixes n :: nat and p :: real assumes p_nonneg: "0 \<le> p" and p_le_1: "p \<le> 1" |
|
604 |
begin |
|
605 |
||
606 |
lift_definition binomial_pmf :: "nat pmf" is "\<lambda>k. (n choose k) * p^k * (1 - p)^(n - k)" |
|
607 |
proof |
|
608 |
have "(\<integral>\<^sup>+k. ereal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) \<partial>count_space UNIV) = |
|
609 |
ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))" |
|
610 |
using p_le_1 p_nonneg by (subst nn_integral_count_space') auto |
|
611 |
also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n" |
|
612 |
by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def) |
|
613 |
finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1" |
|
614 |
by simp |
|
615 |
qed (insert p_nonneg p_le_1, simp) |
|
616 |
||
617 |
lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)" |
|
618 |
by transfer rule |
|
619 |
||
620 |
lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})" |
|
621 |
using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff) |
|
622 |
||
623 |
end |
|
624 |
||
625 |
end |
|
626 |
||
627 |
lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}" |
|
628 |
by (simp add: set_pmf_binomial_eq) |
|
629 |
||
630 |
lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}" |
|
631 |
by (simp add: set_pmf_binomial_eq) |
|
632 |
||
633 |
lemma set_pmf_binomial[simp]: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (binomial_pmf n p) = {..n}" |
|
634 |
by (simp add: set_pmf_binomial_eq) |
|
635 |
||
636 |
subsection \<open> Monad Interpretation \<close> |
|
59000 | 637 |
|
638 |
lemma measurable_measure_pmf[measurable]: |
|
639 |
"(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))" |
|
640 |
by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales |
|
641 |
||
642 |
lemma bind_pmf_cong: |
|
643 |
assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)" |
|
644 |
assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" |
|
645 |
shows "bind (measure_pmf x) A = bind (measure_pmf x) B" |
|
646 |
proof (rule measure_eqI) |
|
647 |
show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)" |
|
59048 | 648 |
using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra) |
59000 | 649 |
next |
650 |
fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)" |
|
651 |
then have X: "X \<in> sets N" |
|
59048 | 652 |
using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra) |
59000 | 653 |
show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X" |
654 |
using assms |
|
655 |
by (subst (1 2) emeasure_bind[where N=N, OF _ _ X]) |
|
656 |
(auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
657 |
qed |
|
658 |
||
659 |
context |
|
660 |
begin |
|
661 |
||
662 |
interpretation pmf_as_measure . |
|
663 |
||
664 |
lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf" |
|
665 |
proof (intro conjI) |
|
666 |
fix M :: "'a pmf pmf" |
|
667 |
||
668 |
interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf" |
|
59048 | 669 |
apply (intro measure_pmf.prob_space_bind[where S="count_space UNIV"] AE_I2) |
670 |
apply (auto intro!: subprob_space_measure_pmf simp: space_subprob_algebra) |
|
59000 | 671 |
apply unfold_locales |
672 |
done |
|
673 |
show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)" |
|
674 |
by intro_locales |
|
675 |
show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV" |
|
59048 | 676 |
by (subst sets_bind) auto |
59000 | 677 |
have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0" |
59048 | 678 |
by (auto simp: AE_bind[where B="count_space UNIV"] measure_pmf_in_subprob_algebra |
679 |
emeasure_bind[where N="count_space UNIV"] AE_measure_pmf_iff nn_integral_0_iff_AE |
|
680 |
measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq) |
|
59000 | 681 |
then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0" |
682 |
unfolding bind.emeasure_eq_measure by simp |
|
683 |
qed |
|
684 |
||
685 |
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)" |
|
686 |
proof (transfer fixing: N i) |
|
687 |
have N: "subprob_space (measure_pmf N)" |
|
688 |
by (rule prob_space_imp_subprob_space) intro_locales |
|
689 |
show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})" |
|
690 |
using measurable_measure_pmf[of "\<lambda>x. x"] |
|
691 |
by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto |
|
692 |
qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def) |
|
693 |
||
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
694 |
lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
695 |
unfolding pmf_join |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
696 |
by (intro nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
697 |
(auto simp: pmf_le_1 pmf_nonneg) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
698 |
|
59024 | 699 |
lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)" |
700 |
apply(simp add: set_eq_iff set_pmf_iff pmf_join) |
|
701 |
apply(subst integral_nonneg_eq_0_iff_AE) |
|
702 |
apply(auto simp add: pmf_le_1 pmf_nonneg AE_measure_pmf_iff intro!: measure_pmf.integrable_const_bound[where B=1]) |
|
703 |
done |
|
704 |
||
59000 | 705 |
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)" |
706 |
by (auto intro!: prob_space_return simp: AE_return measure_return) |
|
707 |
||
708 |
lemma join_return_pmf: "join_pmf (return_pmf M) = M" |
|
709 |
by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq) |
|
710 |
||
711 |
lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)" |
|
712 |
by transfer (simp add: distr_return) |
|
713 |
||
59052 | 714 |
lemma map_pmf_const[simp]: "map_pmf (\<lambda>_. c) M = return_pmf c" |
715 |
by transfer (auto simp: prob_space.distr_const) |
|
716 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
717 |
lemma set_return_pmf: "set_pmf (return_pmf x) = {x}" |
59000 | 718 |
by transfer (auto simp add: measure_return split: split_indicator) |
719 |
||
720 |
lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x" |
|
721 |
by transfer (simp add: measure_return) |
|
722 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
723 |
lemma nn_integral_return_pmf[simp]: "0 \<le> f x \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>return_pmf x) = f x" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
724 |
unfolding return_pmf.rep_eq by (intro nn_integral_return) auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
725 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
726 |
lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
727 |
unfolding return_pmf.rep_eq by (intro emeasure_return) auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
728 |
|
59000 | 729 |
end |
730 |
||
59475 | 731 |
lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y \<longleftrightarrow> x = y" |
732 |
by (metis insertI1 set_return_pmf singletonD) |
|
733 |
||
59000 | 734 |
definition "bind_pmf M f = join_pmf (map_pmf f M)" |
735 |
||
736 |
lemma (in pmf_as_measure) bind_transfer[transfer_rule]: |
|
737 |
"rel_fun pmf_as_measure.cr_pmf (rel_fun (rel_fun op = pmf_as_measure.cr_pmf) pmf_as_measure.cr_pmf) op \<guillemotright>= bind_pmf" |
|
738 |
proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq) |
|
739 |
fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)" |
|
740 |
then have f: "f = (\<lambda>x. measure_pmf (g x))" |
|
741 |
by auto |
|
742 |
show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf" |
|
743 |
unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto |
|
744 |
qed |
|
745 |
||
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
746 |
lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
747 |
by (auto intro!: nn_integral_distr simp: bind_pmf_def ereal_pmf_join map_pmf.rep_eq) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
748 |
|
59000 | 749 |
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)" |
750 |
by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq) |
|
751 |
||
752 |
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x" |
|
753 |
unfolding bind_pmf_def map_return_pmf join_return_pmf .. |
|
754 |
||
59052 | 755 |
lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id" |
756 |
by (simp add: bind_pmf_def) |
|
757 |
||
758 |
lemma bind_pmf_const[simp]: "bind_pmf M (\<lambda>x. c) = c" |
|
759 |
unfolding bind_pmf_def map_pmf_const join_return_pmf .. |
|
760 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
761 |
lemma set_bind_pmf: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
762 |
apply (simp add: set_eq_iff set_pmf_iff pmf_bind) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
763 |
apply (subst integral_nonneg_eq_0_iff_AE) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
764 |
apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
765 |
intro!: measure_pmf.integrable_const_bound[where B=1]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
766 |
done |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
767 |
|
59425 | 768 |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
769 |
lemma measurable_pair_restrict_pmf2: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
770 |
assumes "countable A" |
59425 | 771 |
assumes [measurable]: "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> measurable M L" |
772 |
shows "f \<in> measurable (M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A) L" (is "f \<in> measurable ?M _") |
|
773 |
proof - |
|
774 |
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
|
775 |
by (simp add: restrict_count_space) |
|
776 |
||
777 |
show ?thesis |
|
778 |
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (fst b, a)" and g=snd and I=A, |
|
779 |
unfolded pair_collapse] assms) |
|
780 |
measurable |
|
781 |
qed |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
782 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
783 |
lemma measurable_pair_restrict_pmf1: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
784 |
assumes "countable A" |
59425 | 785 |
assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N L" |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
786 |
shows "f \<in> measurable (restrict_space (measure_pmf M) A \<Otimes>\<^sub>M N) L" |
59425 | 787 |
proof - |
788 |
have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" |
|
789 |
by (simp add: restrict_count_space) |
|
790 |
||
791 |
show ?thesis |
|
792 |
by (intro measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, |
|
793 |
unfolded pair_collapse] assms) |
|
794 |
measurable |
|
795 |
qed |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
796 |
|
59000 | 797 |
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))" |
798 |
unfolding pmf_eq_iff pmf_bind |
|
799 |
proof |
|
800 |
fix i |
|
801 |
interpret B: prob_space "restrict_space B B" |
|
802 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
803 |
(auto simp: AE_measure_pmf_iff) |
|
804 |
interpret A: prob_space "restrict_space A A" |
|
805 |
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) |
|
806 |
(auto simp: AE_measure_pmf_iff) |
|
807 |
||
808 |
interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B" |
|
809 |
by unfold_locales |
|
810 |
||
811 |
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)" |
|
812 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict) |
|
813 |
also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)" |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
814 |
by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
815 |
countable_set_pmf borel_measurable_count_space) |
59000 | 816 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)" |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
817 |
by (rule AB.Fubini_integral[symmetric]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
818 |
(auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2 |
59023 | 819 |
simp: pmf_nonneg pmf_le_1 measurable_restrict_space1) |
59000 | 820 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)" |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
821 |
by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
822 |
countable_set_pmf borel_measurable_count_space) |
59000 | 823 |
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)" |
824 |
by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric]) |
|
825 |
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)" . |
|
826 |
qed |
|
827 |
||
828 |
||
829 |
context |
|
830 |
begin |
|
831 |
||
832 |
interpretation pmf_as_measure . |
|
833 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
834 |
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
835 |
by transfer simp |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
836 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
837 |
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)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
838 |
using measurable_measure_pmf[of N] |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
839 |
unfolding measure_pmf_bind |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
840 |
apply (subst (1 3) nn_integral_max_0[symmetric]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
841 |
apply (intro nn_integral_bind[where B="count_space UNIV"]) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
842 |
apply auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
843 |
done |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
844 |
|
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
845 |
lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (\<integral>\<^sup>+x. emeasure (N x) X \<partial>M)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
846 |
using measurable_measure_pmf[of N] |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
847 |
unfolding measure_pmf_bind |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
848 |
by (subst emeasure_bind[where N="count_space UNIV"]) auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
849 |
|
59000 | 850 |
lemma bind_return_pmf': "bind_pmf N return_pmf = N" |
851 |
proof (transfer, clarify) |
|
852 |
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N" |
|
853 |
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return') |
|
854 |
qed |
|
855 |
||
856 |
lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N" |
|
857 |
proof (transfer, clarify) |
|
858 |
fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV" |
|
859 |
then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f" |
|
860 |
by (subst bind_return_distr[symmetric]) |
|
861 |
(auto simp: prob_space.not_empty measurable_def comp_def) |
|
862 |
qed |
|
863 |
||
864 |
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)" |
|
865 |
by transfer |
|
866 |
(auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"] |
|
867 |
simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space) |
|
868 |
||
869 |
end |
|
870 |
||
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
871 |
lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
872 |
unfolding bind_return_pmf''[symmetric] bind_assoc_pmf[of M] .. |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
873 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
874 |
lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
875 |
unfolding bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf .. |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
876 |
|
59052 | 877 |
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)" |
878 |
unfolding bind_pmf_def[symmetric] |
|
879 |
unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf |
|
880 |
by (simp add: bind_return_pmf'') |
|
881 |
||
59000 | 882 |
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))" |
883 |
||
884 |
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b" |
|
885 |
unfolding pair_pmf_def pmf_bind pmf_return |
|
886 |
apply (subst integral_measure_pmf[where A="{b}"]) |
|
887 |
apply (auto simp: indicator_eq_0_iff) |
|
888 |
apply (subst integral_measure_pmf[where A="{a}"]) |
|
889 |
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg) |
|
890 |
done |
|
891 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
892 |
lemma set_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
893 |
unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
894 |
|
59048 | 895 |
lemma measure_pmf_in_subprob_space[measurable (raw)]: |
896 |
"measure_pmf M \<in> space (subprob_algebra (count_space UNIV))" |
|
897 |
by (simp add: space_subprob_algebra) intro_locales |
|
898 |
||
59134 | 899 |
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)" |
900 |
proof - |
|
901 |
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)" |
|
902 |
by (subst nn_integral_max_0[symmetric]) |
|
903 |
(auto simp: AE_measure_pmf_iff set_pair_pmf intro!: nn_integral_cong_AE) |
|
904 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) * indicator (A \<times> B) (a, b) \<partial>B \<partial>A)" |
|
905 |
by (simp add: pair_pmf_def) |
|
906 |
also have "\<dots> = (\<integral>\<^sup>+a. \<integral>\<^sup>+b. max 0 (f (a, b)) \<partial>B \<partial>A)" |
|
907 |
by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) |
|
908 |
finally show ?thesis |
|
909 |
unfolding nn_integral_max_0 . |
|
910 |
qed |
|
911 |
||
912 |
lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)" |
|
913 |
proof (safe intro!: pmf_eqI) |
|
914 |
fix a :: "'a" and b :: "'b" |
|
915 |
have [simp]: "\<And>c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ereal)" |
|
916 |
by (auto split: split_indicator) |
|
917 |
||
918 |
have "ereal (pmf (pair_pmf (map_pmf f A) B) (a, b)) = |
|
919 |
ereal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))" |
|
920 |
unfolding pmf_pair ereal_pmf_map |
|
921 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg |
|
922 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
923 |
then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)" |
|
924 |
by simp |
|
925 |
qed |
|
926 |
||
927 |
lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)" |
|
928 |
proof (safe intro!: pmf_eqI) |
|
929 |
fix a :: "'a" and b :: "'b" |
|
930 |
have [simp]: "\<And>c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ereal)" |
|
931 |
by (auto split: split_indicator) |
|
932 |
||
933 |
have "ereal (pmf (pair_pmf A (map_pmf f B)) (a, b)) = |
|
934 |
ereal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))" |
|
935 |
unfolding pmf_pair ereal_pmf_map |
|
936 |
by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg |
|
937 |
emeasure_map_pmf[symmetric] del: emeasure_map_pmf) |
|
938 |
then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)" |
|
939 |
by simp |
|
940 |
qed |
|
941 |
||
942 |
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)" |
|
943 |
by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta') |
|
944 |
||
59000 | 945 |
lemma bind_pair_pmf: |
946 |
assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)" |
|
947 |
shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))" |
|
948 |
(is "?L = ?R") |
|
949 |
proof (rule measure_eqI) |
|
950 |
have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)" |
|
951 |
using M[THEN measurable_space] by (simp_all add: space_pair_measure) |
|
952 |
||
59048 | 953 |
note measurable_bind[where N="count_space UNIV", measurable] |
954 |
note measure_pmf_in_subprob_space[simp] |
|
955 |
||
59000 | 956 |
have sets_eq_N: "sets ?L = N" |
59048 | 957 |
by (subst sets_bind[OF sets_kernel[OF M']]) auto |
59000 | 958 |
show "sets ?L = sets ?R" |
59048 | 959 |
using measurable_space[OF M] |
960 |
by (simp add: sets_eq_N space_pair_measure space_subprob_algebra) |
|
59000 | 961 |
fix X assume "X \<in> sets ?L" |
962 |
then have X[measurable]: "X \<in> sets N" |
|
963 |
unfolding sets_eq_N . |
|
964 |
then show "emeasure ?L X = emeasure ?R X" |
|
965 |
apply (simp add: emeasure_bind[OF _ M' X]) |
|
59048 | 966 |
apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A] |
967 |
nn_integral_measure_pmf_finite set_return_pmf emeasure_nonneg pmf_return one_ereal_def[symmetric]) |
|
968 |
apply (subst emeasure_bind[OF _ _ X]) |
|
59000 | 969 |
apply measurable |
970 |
apply (subst emeasure_bind[OF _ _ X]) |
|
971 |
apply measurable |
|
972 |
done |
|
973 |
qed |
|
974 |
||
59052 | 975 |
lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A" |
976 |
unfolding bind_pmf_def[symmetric] bind_return_pmf' .. |
|
977 |
||
978 |
lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A" |
|
979 |
by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
980 |
||
981 |
lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B" |
|
982 |
by (simp add: pair_pmf_def bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf bind_return_pmf') |
|
983 |
||
59053 | 984 |
lemma nn_integral_pmf': |
985 |
"inj_on f A \<Longrightarrow> (\<integral>\<^sup>+x. pmf p (f x) \<partial>count_space A) = emeasure p (f ` A)" |
|
986 |
by (subst nn_integral_bij_count_space[where g=f and B="f`A"]) |
|
987 |
(auto simp: bij_betw_def nn_integral_pmf) |
|
988 |
||
989 |
lemma pmf_le_0_iff[simp]: "pmf M p \<le> 0 \<longleftrightarrow> pmf M p = 0" |
|
990 |
using pmf_nonneg[of M p] by simp |
|
991 |
||
992 |
lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0" |
|
993 |
using pmf_nonneg[of M p] by simp_all |
|
994 |
||
995 |
lemma pmf_eq_0_set_pmf: "pmf M p = 0 \<longleftrightarrow> p \<notin> set_pmf M" |
|
996 |
unfolding set_pmf_iff by simp |
|
997 |
||
998 |
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" |
|
999 |
by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD |
|
1000 |
intro!: measure_pmf.finite_measure_eq_AE) |
|
1001 |
||
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1002 |
subsection \<open> Conditional Probabilities \<close> |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1003 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1004 |
context |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1005 |
fixes p :: "'a pmf" and s :: "'a set" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1006 |
assumes not_empty: "set_pmf p \<inter> s \<noteq> {}" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1007 |
begin |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1008 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1009 |
interpretation pmf_as_measure . |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1010 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1011 |
lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1012 |
proof |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1013 |
assume "emeasure (measure_pmf p) s = 0" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1014 |
then have "AE x in measure_pmf p. x \<notin> s" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1015 |
by (rule AE_I[rotated]) auto |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1016 |
with not_empty show False |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1017 |
by (auto simp: AE_measure_pmf_iff) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1018 |
qed |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1019 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1020 |
lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1021 |
using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1022 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1023 |
lift_definition cond_pmf :: "'a pmf" is |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1024 |
"uniform_measure (measure_pmf p) s" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1025 |
proof (intro conjI) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1026 |
show "prob_space (uniform_measure (measure_pmf p) s)" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1027 |
by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1028 |
show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1029 |
by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1030 |
AE_measure_pmf_iff set_pmf.rep_eq) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1031 |
qed simp |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1032 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1033 |
lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1034 |
by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1035 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1036 |
lemma set_cond_pmf: "set_pmf cond_pmf = set_pmf p \<inter> s" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1037 |
by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1038 |
|
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1039 |
end |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1040 |
|
59494 | 1041 |
lemma cond_map_pmf: |
1042 |
assumes "set_pmf p \<inter> f -` s \<noteq> {}" |
|
1043 |
shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))" |
|
1044 |
proof - |
|
1045 |
have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}" |
|
1046 |
using assms by (simp add: set_map_pmf) auto |
|
1047 |
{ fix x |
|
1048 |
have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) = |
|
1049 |
emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)" |
|
1050 |
unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure) |
|
1051 |
also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})" |
|
1052 |
by auto |
|
1053 |
also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) = |
|
1054 |
ereal (pmf (cond_pmf (map_pmf f p) s) x)" |
|
1055 |
using measure_measure_pmf_not_zero[OF *] |
|
1056 |
by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric] |
|
1057 |
del: ereal_divide) |
|
1058 |
finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)" |
|
1059 |
by simp } |
|
1060 |
then show ?thesis |
|
1061 |
by (intro pmf_eqI) simp |
|
1062 |
qed |
|
1063 |
||
59495 | 1064 |
lemma bind_cond_pmf_cancel: |
1065 |
assumes in_S: "\<And>x. x \<in> set_pmf p \<Longrightarrow> x \<in> S x" |
|
1066 |
assumes S_eq: "\<And>x y. x \<in> S y \<Longrightarrow> S x = S y" |
|
1067 |
shows "bind_pmf p (\<lambda>x. cond_pmf p (S x)) = p" |
|
1068 |
proof (rule pmf_eqI) |
|
1069 |
have [simp]: "\<And>x. x \<in> p \<Longrightarrow> p \<inter> (S x) \<noteq> {}" |
|
1070 |
using in_S by auto |
|
1071 |
fix z |
|
1072 |
have pmf_le: "pmf p z \<le> measure p (S z)" |
|
1073 |
proof cases |
|
1074 |
assume "z \<in> p" from in_S[OF this] show ?thesis |
|
1075 |
by (auto intro!: measure_pmf.finite_measure_mono simp: pmf.rep_eq) |
|
1076 |
qed (simp add: set_pmf_iff measure_nonneg) |
|
1077 |
||
1078 |
have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z) = |
|
1079 |
(\<integral>\<^sup>+ x. ereal (pmf p z / measure p (S z)) * indicator (S z) x \<partial>p)" |
|
1080 |
by (subst ereal_pmf_bind) |
|
1081 |
(auto intro!: nn_integral_cong_AE dest!: S_eq split: split_indicator |
|
1082 |
simp: AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf in_S) |
|
1083 |
also have "\<dots> = pmf p z" |
|
1084 |
using pmf_le pmf_nonneg[of p z] |
|
1085 |
by (subst nn_integral_cmult) (simp_all add: measure_nonneg measure_pmf.emeasure_eq_measure) |
|
1086 |
finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z = pmf p z" |
|
1087 |
by simp |
|
1088 |
qed |
|
1089 |
||
59023 | 1090 |
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool" |
1091 |
for R p q |
|
1092 |
where |
|
1093 |
"\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y; |
|
1094 |
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk> |
|
1095 |
\<Longrightarrow> rel_pmf R p q" |
|
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1096 |
|
59023 | 1097 |
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1098 |
proof - |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1099 |
show "map_pmf id = id" by (rule map_pmf_id) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1100 |
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1101 |
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" |
59023 | 1102 |
by (intro map_pmf_cong refl) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1103 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1104 |
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1105 |
by (rule pmf_set_map) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1106 |
|
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1107 |
{ fix p :: "'s pmf" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1108 |
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1109 |
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) |
59053 | 1110 |
(auto intro: countable_set_pmf) |
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1111 |
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq" |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1112 |
by (metis Field_natLeq card_of_least natLeq_Well_order) |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1113 |
finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . } |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1114 |
|
59023 | 1115 |
show "\<And>R. rel_pmf R = |
1116 |
(BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO |
|
1117 |
BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)" |
|
1118 |
by (auto simp add: fun_eq_iff BNF_Def.Grp_def OO_def rel_pmf.simps) |
|
1119 |
||
1120 |
{ fix p :: "'a pmf" and f :: "'a \<Rightarrow> 'b" and g x |
|
1121 |
assume p: "\<And>z. z \<in> set_pmf p \<Longrightarrow> f z = g z" |
|
1122 |
and x: "x \<in> set_pmf p" |
|
1123 |
thus "f x = g x" by simp } |
|
1124 |
||
1125 |
fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" |
|
1126 |
{ fix p q r |
|
1127 |
assume pq: "rel_pmf R p q" |
|
1128 |
and qr:"rel_pmf S q r" |
|
1129 |
from pq obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" |
|
1130 |
and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto |
|
1131 |
from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z" |
|
1132 |
and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto |
|
1133 |
||
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1134 |
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)))" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1135 |
have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {(y', z). y' = y} \<noteq> {}" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1136 |
by (force simp: q' set_map_pmf) |
59023 | 1137 |
|
59053 | 1138 |
have "rel_pmf (R OO S) p r" |
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1139 |
proof (rule rel_pmf.intros) |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1140 |
fix x z assume "(x, z) \<in> pr" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1141 |
then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr" |
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1142 |
by (auto simp: q pr_welldefined pr_def set_bind_pmf split_beta set_return_pmf set_cond_pmf set_map_pmf) |
59053 | 1143 |
with pq qr show "(R OO S) x z" |
1144 |
by blast |
|
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1145 |
next |
59495 | 1146 |
have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {(y', z). y' = y}))" |
1147 |
by (simp add: pr_def q split_beta bind_map_pmf bind_return_pmf'' map_bind_pmf map_return_pmf) |
|
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1148 |
then show "map_pmf snd pr = r" |
59495 | 1149 |
unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) auto |
59493
e088f1b2f2fc
introduce discrete conditional probabilities, use it to simplify bnf proof of pmf
hoelzl
parents:
59475
diff
changeset
|
1150 |
qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf bind_return_pmf'' p) } |
59023 | 1151 |
then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)" |
1152 |
by(auto simp add: le_fun_def) |
|
1153 |
qed (fact natLeq_card_order natLeq_cinfinite)+ |
|
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1154 |
|
59134 | 1155 |
lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M \<longleftrightarrow> (\<forall>a\<in>M. R x a)" |
1156 |
proof safe |
|
1157 |
fix a assume "a \<in> M" "rel_pmf R (return_pmf x) M" |
|
1158 |
then obtain pq where *: "\<And>a b. (a, b) \<in> set_pmf pq \<Longrightarrow> R a b" |
|
1159 |
and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq" |
|
1160 |
by (force elim: rel_pmf.cases) |
|
1161 |
moreover have "set_pmf (return_pmf x) = {x}" |
|
1162 |
by (simp add: set_return_pmf) |
|
1163 |
with `a \<in> M` have "(x, a) \<in> pq" |
|
1164 |
by (force simp: eq set_map_pmf) |
|
1165 |
with * show "R x a" |
|
1166 |
by auto |
|
1167 |
qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"] |
|
1168 |
simp: map_fst_pair_pmf map_snd_pair_pmf set_pair_pmf set_return_pmf) |
|
1169 |
||
1170 |
lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) \<longleftrightarrow> (\<forall>a\<in>M. R a x)" |
|
1171 |
by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1) |
|
1172 |
||
59475 | 1173 |
lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2" |
1174 |
unfolding rel_pmf_return_pmf2 set_return_pmf by simp |
|
1175 |
||
1176 |
lemma rel_pmf_False[simp]: "rel_pmf (\<lambda>x y. False) x y = False" |
|
1177 |
unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce |
|
1178 |
||
59134 | 1179 |
lemma rel_pmf_rel_prod: |
1180 |
"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'" |
|
1181 |
proof safe |
|
1182 |
assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
1183 |
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" |
|
1184 |
and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'" |
|
1185 |
by (force elim: rel_pmf.cases) |
|
1186 |
show "rel_pmf R A B" |
|
1187 |
proof (rule rel_pmf.intros) |
|
1188 |
let ?f = "\<lambda>(a, b). (fst a, fst b)" |
|
1189 |
have [simp]: "(\<lambda>x. fst (?f x)) = fst o fst" "(\<lambda>x. snd (?f x)) = fst o snd" |
|
1190 |
by auto |
|
1191 |
||
1192 |
show "map_pmf fst (map_pmf ?f pq) = A" |
|
1193 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1194 |
show "map_pmf snd (map_pmf ?f pq) = B" |
|
1195 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) |
|
1196 |
||
1197 |
fix a b assume "(a, b) \<in> set_pmf (map_pmf ?f pq)" |
|
1198 |
then obtain c d where "((a, c), (b, d)) \<in> set_pmf pq" |
|
1199 |
by (auto simp: set_map_pmf) |
|
1200 |
from pq[OF this] show "R a b" .. |
|
1201 |
qed |
|
1202 |
show "rel_pmf S A' B'" |
|
1203 |
proof (rule rel_pmf.intros) |
|
1204 |
let ?f = "\<lambda>(a, b). (snd a, snd b)" |
|
1205 |
have [simp]: "(\<lambda>x. fst (?f x)) = snd o fst" "(\<lambda>x. snd (?f x)) = snd o snd" |
|
1206 |
by auto |
|
1207 |
||
1208 |
show "map_pmf fst (map_pmf ?f pq) = A'" |
|
1209 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1210 |
show "map_pmf snd (map_pmf ?f pq) = B'" |
|
1211 |
by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) |
|
1212 |
||
1213 |
fix c d assume "(c, d) \<in> set_pmf (map_pmf ?f pq)" |
|
1214 |
then obtain a b where "((a, c), (b, d)) \<in> set_pmf pq" |
|
1215 |
by (auto simp: set_map_pmf) |
|
1216 |
from pq[OF this] show "S c d" .. |
|
1217 |
qed |
|
1218 |
next |
|
1219 |
assume "rel_pmf R A B" "rel_pmf S A' B'" |
|
1220 |
then obtain Rpq Spq |
|
1221 |
where Rpq: "\<And>a b. (a, b) \<in> set_pmf Rpq \<Longrightarrow> R a b" |
|
1222 |
"map_pmf fst Rpq = A" "map_pmf snd Rpq = B" |
|
1223 |
and Spq: "\<And>a b. (a, b) \<in> set_pmf Spq \<Longrightarrow> S a b" |
|
1224 |
"map_pmf fst Spq = A'" "map_pmf snd Spq = B'" |
|
1225 |
by (force elim: rel_pmf.cases) |
|
1226 |
||
1227 |
let ?f = "(\<lambda>((a, c), (b, d)). ((a, b), (c, d)))" |
|
1228 |
let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)" |
|
1229 |
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))" |
|
1230 |
by auto |
|
1231 |
||
1232 |
show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" |
|
1233 |
by (rule rel_pmf.intros[where pq="?pq"]) |
|
1234 |
(auto simp: map_snd_pair_pmf map_fst_pair_pmf set_pair_pmf set_map_pmf map_pmf_comp Rpq Spq |
|
1235 |
map_pair) |
|
1236 |
qed |
|
1237 |
||
58587
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1238 |
end |
5484f6079bcd
add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff
changeset
|
1239 |