(* Title: HOL/Probability/Probability_Mass_Function.thy
Author: Johannes Hölzl, TU München *)
section \<open> Probability mass function \<close>
theory Probability_Mass_Function
imports
Giry_Monad
"~~/src/HOL/Library/Multiset"
begin
lemma (in finite_measure) countable_support: (* replace version in pmf *)
"countable {x. measure M {x} \<noteq> 0}"
proof cases
assume "measure M (space M) = 0"
with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
by auto
then show ?thesis
by simp
next
let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
assume "?M \<noteq> 0"
then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
using reals_Archimedean[of "?m x / ?M" for x]
by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)
have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
proof (rule ccontr)
fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
by (metis infinite_arbitrarily_large)
from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
by auto
{ fix x assume "x \<in> X"
from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
note singleton_sets = this
have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
using `?M \<noteq> 0`
by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
by (rule setsum_mono) fact
also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
using singleton_sets `finite X`
by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
finally have "?M < measure M (\<Union>x\<in>X. {x})" .
moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
ultimately show False by simp
qed
show ?thesis
unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
qed
lemma (in finite_measure) AE_support_countable:
assumes [simp]: "sets M = UNIV"
shows "(AE x in M. measure M {x} \<noteq> 0) \<longleftrightarrow> (\<exists>S. countable S \<and> (AE x in M. x \<in> S))"
proof
assume "\<exists>S. countable S \<and> (AE x in M. x \<in> S)"
then obtain S where S[intro]: "countable S" and ae: "AE x in M. x \<in> S"
by auto
then have "emeasure M (\<Union>x\<in>{x\<in>S. emeasure M {x} \<noteq> 0}. {x}) =
(\<integral>\<^sup>+ x. emeasure M {x} * indicator {x\<in>S. emeasure M {x} \<noteq> 0} x \<partial>count_space UNIV)"
by (subst emeasure_UN_countable)
(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} * indicator S x \<partial>count_space UNIV)"
by (auto intro!: nn_integral_cong split: split_indicator)
also have "\<dots> = emeasure M (\<Union>x\<in>S. {x})"
by (subst emeasure_UN_countable)
(auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space)
also have "\<dots> = emeasure M (space M)"
using ae by (intro emeasure_eq_AE) auto
finally have "emeasure M {x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0} = emeasure M (space M)"
by (simp add: emeasure_single_in_space cong: rev_conj_cong)
with finite_measure_compl[of "{x \<in> space M. x\<in>S \<and> emeasure M {x} \<noteq> 0}"]
have "AE x in M. x \<in> S \<and> emeasure M {x} \<noteq> 0"
by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure set_diff_eq cong: conj_cong)
then show "AE x in M. measure M {x} \<noteq> 0"
by (auto simp: emeasure_eq_measure)
qed (auto intro!: exI[of _ "{x. measure M {x} \<noteq> 0}"] countable_support)
subsection {* PMF as measure *}
typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
morphisms measure_pmf Abs_pmf
by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
(auto intro!: prob_space_uniform_measure AE_uniform_measureI)
declare [[coercion measure_pmf]]
lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
using pmf.measure_pmf[of p] by auto
interpretation measure_pmf!: prob_space "measure_pmf M" for M
by (rule prob_space_measure_pmf)
interpretation measure_pmf!: subprob_space "measure_pmf M" for M
by (rule prob_space_imp_subprob_space) unfold_locales
locale pmf_as_measure
begin
setup_lifting type_definition_pmf
end
context
begin
interpretation pmf_as_measure .
lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
"\<lambda>f M. distr M (count_space UNIV) f"
proof safe
fix M and f :: "'a \<Rightarrow> 'b"
let ?D = "distr M (count_space UNIV) f"
assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
interpret prob_space M by fact
from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
proof eventually_elim
fix x
have "measure M {x} \<le> measure M (f -` {f x})"
by (intro finite_measure_mono) auto
then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
using measure_nonneg[of M "{x}"] by auto
qed
then show "AE x in ?D. measure ?D {x} \<noteq> 0"
by (simp add: AE_distr_iff measure_distr measurable_def)
qed (auto simp: measurable_def prob_space.prob_space_distr)
declare [[coercion set_pmf]]
lemma countable_set_pmf: "countable (set_pmf p)"
by transfer (metis prob_space.finite_measure finite_measure.countable_support)
lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
by transfer metis
lemma sets_measure_pmf_count_space: "sets (measure_pmf M) = sets (count_space UNIV)"
by simp
lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
by (auto simp: measurable_def)
lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
by (intro measurable_cong_sets) simp_all
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
by transfer (simp add: less_le measure_nonneg)
lemma pmf_nonneg: "0 \<le> pmf p x"
by transfer (simp add: measure_nonneg)
lemma pmf_le_1: "pmf p x \<le> 1"
by (simp add: pmf.rep_eq)
lemma emeasure_pmf_single:
fixes M :: "'a pmf"
shows "emeasure M {x} = pmf M x"
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
by transfer simp
lemma emeasure_pmf_single_eq_zero_iff:
fixes M :: "'a pmf"
shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
proof -
{ fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
with P have "AE x in M. x \<noteq> y"
by auto
with y have False
by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
then show ?thesis
using AE_measure_pmf[of M] by auto
qed
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
using AE_measure_pmf[of M] by (intro notI) simp
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
by transfer simp
lemma emeasure_measure_pmf_finite: "finite S \<Longrightarrow> emeasure (measure_pmf M) S = (\<Sum>s\<in>S. pmf M s)"
by (subst emeasure_eq_setsum_singleton) (auto simp: emeasure_pmf_single)
lemma nn_integral_measure_pmf_support:
fixes f :: "'a \<Rightarrow> ereal"
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"
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>A. f x * pmf M x)"
proof -
have "(\<integral>\<^sup>+x. f x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator)
also have "\<dots> = (\<Sum>x\<in>A. f x * emeasure M {x})"
using assms by (intro nn_integral_indicator_finite) auto
finally show ?thesis
by (simp add: emeasure_measure_pmf_finite)
qed
lemma nn_integral_measure_pmf_finite:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "finite (set_pmf M)" and nn: "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x"
shows "(\<integral>\<^sup>+x. f x \<partial>measure_pmf M) = (\<Sum>x\<in>set_pmf M. f x * pmf M x)"
using assms by (intro nn_integral_measure_pmf_support) auto
lemma integrable_measure_pmf_finite:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite)
lemma integral_measure_pmf:
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
proof -
have "(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x * indicator A x \<partial>measure_pmf M)"
using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff)
also have "\<dots> = (\<Sum>a\<in>A. f a * pmf M a)"
by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite)
finally show ?thesis .
qed
lemma integrable_pmf: "integrable (count_space X) (pmf M)"
proof -
have " (\<integral>\<^sup>+ x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+ x. pmf M x \<partial>count_space (M \<inter> X))"
by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator)
then have "integrable (count_space X) (pmf M) = integrable (count_space (M \<inter> X)) (pmf M)"
by (simp add: integrable_iff_bounded pmf_nonneg)
then show ?thesis
by (simp add: pmf.rep_eq measure_pmf.integrable_measure countable_set_pmf disjoint_family_on_def)
qed
lemma integral_pmf: "(\<integral>x. pmf M x \<partial>count_space X) = measure M X"
proof -
have "(\<integral>x. pmf M x \<partial>count_space X) = (\<integral>\<^sup>+x. pmf M x \<partial>count_space X)"
by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral)
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space (X \<inter> M))"
by (auto intro!: nn_integral_cong_AE split: split_indicator
simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator
AE_count_space set_pmf_iff)
also have "\<dots> = emeasure M (X \<inter> M)"
by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf)
also have "\<dots> = emeasure M X"
by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff)
finally show ?thesis
by (simp add: measure_pmf.emeasure_eq_measure)
qed
lemma integral_pmf_restrict:
"(f::'a \<Rightarrow> 'b::{banach, second_countable_topology}) \<in> borel_measurable (count_space UNIV) \<Longrightarrow>
(\<integral>x. f x \<partial>measure_pmf M) = (\<integral>x. f x \<partial>restrict_space M M)"
by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff)
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
proof -
have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
then show ?thesis
using measure_pmf.emeasure_space_1 by simp
qed
lemma map_pmf_id[simp]: "map_pmf id = id"
by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
using map_pmf_compose[of f g] by (simp add: comp_def)
lemma map_pmf_cong:
assumes "p = q"
shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
lemma pmf_set_map:
fixes f :: "'a \<Rightarrow> 'b"
shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
interpret prob_space M by fact
show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
proof safe
fix x assume "measure M (f -` {x}) \<noteq> 0"
moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
using ae by (intro finite_measure_eq_AE) auto
ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
by (metis measure_empty)
then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
by auto
next
fix x assume "measure M {x} \<noteq> 0"
then have "0 < measure M {x}"
using measure_nonneg[of M "{x}"] by auto
also have "measure M {x} \<le> measure M (f -` {f x})"
by (intro finite_measure_mono) auto
finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
by simp
qed
qed
lemma set_map_pmf: "set_pmf (map_pmf f M) = f`set_pmf M"
using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff)
subsection {* PMFs as function *}
context
fixes f :: "'a \<Rightarrow> real"
assumes nonneg: "\<And>x. 0 \<le> f x"
assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
begin
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
proof (intro conjI)
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
by (simp split: split_indicator)
show "AE x in density (count_space UNIV) (ereal \<circ> f).
measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator)
show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
by default (simp add: emeasure_density prob)
qed simp
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
proof transfer
have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
by (simp split: split_indicator)
fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg)
qed
end
lemma embed_pmf_transfer:
"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"
by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
proof (transfer, elim conjE)
fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
assume "prob_space M" then interpret prob_space M .
show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
proof (rule measure_eqI)
fix A :: "'a set"
have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
(\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
(auto simp: disjoint_family_on_def)
also have "\<dots> = emeasure M A"
using ae by (intro emeasure_eq_AE) auto
finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
using emeasure_space_1 by (simp add: emeasure_density)
qed simp
qed
lemma td_pmf_embed_pmf:
"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}"
unfolding type_definition_def
proof safe
fix p :: "'a pmf"
have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
using measure_pmf.emeasure_space_1[of p] by simp
then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
show "embed_pmf (pmf p) = p"
by (intro measure_pmf_inject[THEN iffD1])
(simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
next
fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
then show "pmf (embed_pmf f) = f"
by (auto intro!: pmf_embed_pmf)
qed (rule pmf_nonneg)
end
locale pmf_as_function
begin
setup_lifting td_pmf_embed_pmf
lemma set_pmf_transfer[transfer_rule]:
assumes "bi_total A"
shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
using `bi_total A`
by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
metis+
end
context
begin
interpretation pmf_as_function .
lemma pmf_eqI: "(\<And>i. pmf M i = pmf N i) \<Longrightarrow> M = N"
by transfer auto
lemma pmf_eq_iff: "M = N \<longleftrightarrow> (\<forall>i. pmf M i = pmf N i)"
by (auto intro: pmf_eqI)
end
context
begin
interpretation pmf_as_function .
lift_definition bernoulli_pmf :: "real \<Rightarrow> bool pmf" is
"\<lambda>p b. ((\<lambda>p. if b then p else 1 - p) \<circ> min 1 \<circ> max 0) p"
by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool
split: split_max split_min)
lemma pmf_bernoulli_True[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) True = p"
by transfer simp
lemma pmf_bernoulli_False[simp]: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> pmf (bernoulli_pmf p) False = 1 - p"
by transfer simp
lemma set_pmf_bernoulli: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (bernoulli_pmf p) = UNIV"
by (auto simp add: set_pmf_iff UNIV_bool)
lift_definition geometric_pmf :: "nat pmf" is "\<lambda>n. 1 / 2^Suc n"
proof
note geometric_sums[of "1 / 2"]
note sums_mult[OF this, of "1 / 2"]
from sums_suminf_ereal[OF this]
show "(\<integral>\<^sup>+ x. ereal (1 / 2 ^ Suc x) \<partial>count_space UNIV) = 1"
by (simp add: nn_integral_count_space_nat field_simps)
qed simp
lemma pmf_geometric[simp]: "pmf geometric_pmf n = 1 / 2^Suc n"
by transfer rule
lemma set_pmf_geometric: "set_pmf geometric_pmf = UNIV"
by (auto simp: set_pmf_iff)
context
fixes M :: "'a multiset" assumes M_not_empty: "M \<noteq> {#}"
begin
lift_definition pmf_of_multiset :: "'a pmf" is "\<lambda>x. count M x / size M"
proof
show "(\<integral>\<^sup>+ x. ereal (real (count M x) / real (size M)) \<partial>count_space UNIV) = 1"
using M_not_empty
by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size
setsum_divide_distrib[symmetric])
(auto simp: size_multiset_overloaded_eq intro!: setsum.cong)
qed simp
lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M"
by transfer rule
lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_of M"
by (auto simp: set_pmf_iff)
end
context
fixes S :: "'a set" assumes S_not_empty: "S \<noteq> {}" and S_finite: "finite S"
begin
lift_definition pmf_of_set :: "'a pmf" is "\<lambda>x. indicator S x / card S"
proof
show "(\<integral>\<^sup>+ x. ereal (indicator S x / real (card S)) \<partial>count_space UNIV) = 1"
using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) auto
qed simp
lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S"
by transfer rule
lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S"
using S_finite S_not_empty by (auto simp: set_pmf_iff)
end
end
subsection {* Monad interpretation *}
lemma measurable_measure_pmf[measurable]:
"(\<lambda>x. measure_pmf (M x)) \<in> measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales
lemma bind_pmf_cong:
assumes "\<And>x. A x \<in> space (subprob_algebra N)" "\<And>x. B x \<in> space (subprob_algebra N)"
assumes "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i"
shows "bind (measure_pmf x) A = bind (measure_pmf x) B"
proof (rule measure_eqI)
show "sets (measure_pmf x \<guillemotright>= A) = sets (measure_pmf x \<guillemotright>= B)"
using assms by (subst (1 2) sets_bind) auto
next
fix X assume "X \<in> sets (measure_pmf x \<guillemotright>= A)"
then have X: "X \<in> sets N"
using assms by (subst (asm) sets_bind) auto
show "emeasure (measure_pmf x \<guillemotright>= A) X = emeasure (measure_pmf x \<guillemotright>= B) X"
using assms
by (subst (1 2) emeasure_bind[where N=N, OF _ _ X])
(auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff)
qed
context
begin
interpretation pmf_as_measure .
lift_definition join_pmf :: "'a pmf pmf \<Rightarrow> 'a pmf" is "\<lambda>M. measure_pmf M \<guillemotright>= measure_pmf"
proof (intro conjI)
fix M :: "'a pmf pmf"
have *: "measure_pmf \<in> measurable (measure_pmf M) (subprob_algebra (count_space UNIV))"
using measurable_measure_pmf[of "\<lambda>x. x"] by simp
interpret bind: prob_space "measure_pmf M \<guillemotright>= measure_pmf"
apply (rule measure_pmf.prob_space_bind[OF _ *])
apply (auto intro!: AE_I2)
apply unfold_locales
done
show "prob_space (measure_pmf M \<guillemotright>= measure_pmf)"
by intro_locales
show "sets (measure_pmf M \<guillemotright>= measure_pmf) = UNIV"
by (subst sets_bind[OF *]) auto
have "AE x in measure_pmf M \<guillemotright>= measure_pmf. emeasure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
by (auto simp add: AE_bind[OF _ *] AE_measure_pmf_iff emeasure_bind[OF _ *]
nn_integral_0_iff_AE measure_pmf.emeasure_eq_measure measure_le_0_iff set_pmf_iff pmf.rep_eq)
then show "AE x in measure_pmf M \<guillemotright>= measure_pmf. measure (measure_pmf M \<guillemotright>= measure_pmf) {x} \<noteq> 0"
unfolding bind.emeasure_eq_measure by simp
qed
lemma pmf_join: "pmf (join_pmf N) i = (\<integral>M. pmf M i \<partial>measure_pmf N)"
proof (transfer fixing: N i)
have N: "subprob_space (measure_pmf N)"
by (rule prob_space_imp_subprob_space) intro_locales
show "measure (measure_pmf N \<guillemotright>= measure_pmf) {i} = integral\<^sup>L (measure_pmf N) (\<lambda>M. measure M {i})"
using measurable_measure_pmf[of "\<lambda>x. x"]
by (intro subprob_space.measure_bind[where N="count_space UNIV", OF N]) auto
qed (auto simp: Transfer.Rel_def rel_fun_def cr_pmf_def)
lift_definition return_pmf :: "'a \<Rightarrow> 'a pmf" is "return (count_space UNIV)"
by (auto intro!: prob_space_return simp: AE_return measure_return)
lemma join_return_pmf: "join_pmf (return_pmf M) = M"
by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq)
lemma map_return_pmf: "map_pmf f (return_pmf x) = return_pmf (f x)"
by transfer (simp add: distr_return)
lemma set_pmf_return: "set_pmf (return_pmf x) = {x}"
by transfer (auto simp add: measure_return split: split_indicator)
lemma pmf_return: "pmf (return_pmf x) y = indicator {y} x"
by transfer (simp add: measure_return)
end
definition "bind_pmf M f = join_pmf (map_pmf f M)"
lemma (in pmf_as_measure) bind_transfer[transfer_rule]:
"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"
proof (auto simp: pmf_as_measure.cr_pmf_def rel_fun_def bind_pmf_def join_pmf.rep_eq map_pmf.rep_eq)
fix M f and g :: "'a \<Rightarrow> 'b pmf" assume "\<forall>x. f x = measure_pmf (g x)"
then have f: "f = (\<lambda>x. measure_pmf (g x))"
by auto
show "measure_pmf M \<guillemotright>= f = distr (measure_pmf M) (count_space UNIV) g \<guillemotright>= measure_pmf"
unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
qed
lemma pmf_bind: "pmf (bind_pmf N f) i = (\<integral>x. pmf (f x) i \<partial>measure_pmf N)"
by (auto intro!: integral_distr simp: bind_pmf_def pmf_join map_pmf.rep_eq)
lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x"
unfolding bind_pmf_def map_return_pmf join_return_pmf ..
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))"
unfolding pmf_eq_iff pmf_bind
proof
fix i
interpret B: prob_space "restrict_space B B"
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
(auto simp: AE_measure_pmf_iff)
interpret A: prob_space "restrict_space A A"
by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE)
(auto simp: AE_measure_pmf_iff)
interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B"
by unfold_locales
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)"
by (rule integral_cong) (auto intro!: integral_pmf_restrict)
also have "\<dots> = (\<integral> x. (\<integral> y. pmf (C x y) i \<partial>restrict_space B B) \<partial>restrict_space A A)"
apply (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral)
apply (auto simp: measurable_split_conv)
apply (subst measurable_cong_sets)
apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
apply (simp add: restrict_count_space)
apply (rule measurable_compose_countable'[OF _ measurable_snd])
apply (rule measurable_compose[OF measurable_fst])
apply (auto intro: countable_set_pmf)
done
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>restrict_space B B)"
apply (rule AB.Fubini_integral[symmetric])
apply (auto intro!: AB.integrable_const_bound[where B=1] simp: pmf_nonneg pmf_le_1)
apply (auto simp: measurable_split_conv)
apply (subst measurable_cong_sets)
apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
apply (simp add: restrict_count_space)
apply (rule measurable_compose_countable'[OF _ measurable_snd])
apply (rule measurable_compose[OF measurable_fst])
apply (auto intro: countable_set_pmf)
done
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>restrict_space A A \<partial>B)"
apply (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral)
apply (auto simp: measurable_split_conv)
apply (subst measurable_cong_sets)
apply (rule sets_pair_measure_cong sets_restrict_space_cong sets_measure_pmf_count_space refl)+
apply (simp add: restrict_count_space)
apply (rule measurable_compose_countable'[OF _ measurable_snd])
apply (rule measurable_compose[OF measurable_fst])
apply (auto intro: countable_set_pmf)
done
also have "\<dots> = (\<integral> y. \<integral> x. pmf (C x y) i \<partial>A \<partial>B)"
by (rule integral_cong) (auto intro!: integral_pmf_restrict[symmetric])
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)" .
qed
context
begin
interpretation pmf_as_measure .
lemma bind_return_pmf': "bind_pmf N return_pmf = N"
proof (transfer, clarify)
fix N :: "'a measure" assume "sets N = UNIV" then show "N \<guillemotright>= return (count_space UNIV) = N"
by (subst return_sets_cong[where N=N]) (simp_all add: bind_return')
qed
lemma bind_return_pmf'': "bind_pmf N (\<lambda>x. return_pmf (f x)) = map_pmf f N"
proof (transfer, clarify)
fix N :: "'b measure" and f :: "'b \<Rightarrow> 'a" assume "prob_space N" "sets N = UNIV"
then show "N \<guillemotright>= (\<lambda>x. return (count_space UNIV) (f x)) = distr N (count_space UNIV) f"
by (subst bind_return_distr[symmetric])
(auto simp: prob_space.not_empty measurable_def comp_def)
qed
lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
by transfer
(auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"]
simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space)
lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M \<guillemotright>= (\<lambda>x. measure_pmf (f x)))"
by transfer simp
end
definition "pair_pmf A B = bind_pmf A (\<lambda>x. bind_pmf B (\<lambda>y. return_pmf (x, y)))"
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
unfolding pair_pmf_def pmf_bind pmf_return
apply (subst integral_measure_pmf[where A="{b}"])
apply (auto simp: indicator_eq_0_iff)
apply (subst integral_measure_pmf[where A="{a}"])
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
done
lemma bind_pair_pmf:
assumes M[measurable]: "M \<in> measurable (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) (subprob_algebra N)"
shows "measure_pmf (pair_pmf A B) \<guillemotright>= M = (measure_pmf A \<guillemotright>= (\<lambda>x. measure_pmf B \<guillemotright>= (\<lambda>y. M (x, y))))"
(is "?L = ?R")
proof (rule measure_eqI)
have M'[measurable]: "M \<in> measurable (pair_pmf A B) (subprob_algebra N)"
using M[THEN measurable_space] by (simp_all add: space_pair_measure)
have sets_eq_N: "sets ?L = N"
by (simp add: sets_bind[OF M'])
show "sets ?L = sets ?R"
unfolding sets_eq_N
apply (subst sets_bind[where N=N])
apply (rule measurable_bind)
apply (rule measurable_compose[OF _ measurable_measure_pmf])
apply measurable
apply (auto intro!: sets_pair_measure_cong sets_measure_pmf_count_space)
done
fix X assume "X \<in> sets ?L"
then have X[measurable]: "X \<in> sets N"
unfolding sets_eq_N .
then show "emeasure ?L X = emeasure ?R X"
apply (simp add: emeasure_bind[OF _ M' X])
unfolding pair_pmf_def measure_pmf_bind[of A]
apply (subst nn_integral_bind[OF _ emeasure_nonneg])
apply (rule measurable_compose[OF M' measurable_emeasure_subprob_algebra, OF X])
apply (subst measurable_cong_sets[OF sets_measure_pmf_count_space refl])
apply (subst subprob_algebra_cong[OF sets_measure_pmf_count_space])
apply measurable
unfolding measure_pmf_bind
apply (subst nn_integral_bind[OF _ emeasure_nonneg])
apply (rule measurable_compose[OF M' measurable_emeasure_subprob_algebra, OF X])
apply (subst measurable_cong_sets[OF sets_measure_pmf_count_space refl])
apply (subst subprob_algebra_cong[OF sets_measure_pmf_count_space])
apply measurable
apply (simp add: nn_integral_measure_pmf_finite set_pmf_return emeasure_nonneg pmf_return one_ereal_def[symmetric])
apply (subst emeasure_bind[OF _ _ X])
apply simp
apply (rule measurable_bind[where N="count_space UNIV"])
apply (rule measurable_compose[OF _ measurable_measure_pmf])
apply measurable
apply (rule sets_pair_measure_cong sets_measure_pmf_count_space refl)+
apply (subst measurable_cong_sets[OF sets_pair_measure_cong[OF sets_measure_pmf_count_space refl] refl])
apply simp
apply (subst emeasure_bind[OF _ _ X])
apply simp
apply (rule measurable_compose[OF _ M])
apply (auto simp: space_pair_measure)
done
qed
lemma set_pmf_bind: "set_pmf (bind_pmf M N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
apply (simp add: set_eq_iff set_pmf_iff pmf_bind)
apply (subst integral_nonneg_eq_0_iff_AE)
apply (auto simp: pmf_nonneg pmf_le_1 AE_measure_pmf_iff
intro!: measure_pmf.integrable_const_bound[where B=1])
done
lemma set_pmf_pair_pmf: "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
unfolding pair_pmf_def set_pmf_bind set_pmf_return by auto
(*
definition
"rel_pmf P d1 d2 \<longleftrightarrow> (\<exists>p3. (\<forall>(x, y) \<in> set_pmf p3. P x y) \<and> map_pmf fst p3 = d1 \<and> map_pmf snd p3 = d2)"
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel
proof -
show "map_pmf id = id" by (rule map_pmf_id)
show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
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"
by (intro map_pmg_cong refl)
show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
by (rule pmf_set_map)
{ fix p :: "'s pmf"
have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
(auto intro: countable_set_pmf inj_on_to_nat_on)
also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
by (metis Field_natLeq card_of_least natLeq_Well_order)
finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
show "\<And>R. pmf_rel R =
(BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def)
{ let ?f = "map_pmf fst" and ?s = "map_pmf snd"
fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and A assume "\<And>x y. (x, y) \<in> set_pmf A \<Longrightarrow> R x y"
fix S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" and B assume "\<And>y z. (y, z) \<in> set_pmf B \<Longrightarrow> S y z"
assume "?f B = ?s A"
have "\<exists>C. (\<forall>(x, z)\<in>set_pmf C. \<exists>y. R x y \<and> S y z) \<and> ?f C = ?f A \<and> ?s C = ?s B"
sorry }
oops
then show "\<And>R::'a \<Rightarrow> 'b \<Rightarrow> bool. \<And>S::'b \<Rightarrow> 'c \<Rightarrow> bool. pmf_rel R OO pmf_rel S \<le> pmf_rel (R OO S)"
by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def)
qed (fact natLeq_card_order natLeq_cinfinite)+
notepad
begin
fix x y :: "nat \<Rightarrow> real"
def IJz \<equiv> "rec_nat ((0, 0), \<lambda>_. 0) (\<lambda>n ((I, J), z).
let a = x I - (\<Sum>j<J. z (I, j)) ; b = y J - (\<Sum>i<I. z (i, J)) in
((if a \<le> b then I + 1 else I, if b \<le> a then J + 1 else J), z((I, J) := min a b)))"
def I == "fst \<circ> fst \<circ> IJz" def J == "snd \<circ> fst \<circ> IJz" def z == "snd \<circ> IJz"
let ?a = "\<lambda>n. x (I n) - (\<Sum>j<J n. z n (I n, j))" and ?b = "\<lambda>n. y (J n) - (\<Sum>i<I n. z n (i, J n))"
have IJz_0[simp]: "\<And>p. z 0 p = 0" "I 0 = 0" "J 0 = 0"
by (simp_all add: I_def J_def z_def IJz_def)
have z_Suc[simp]: "\<And>n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))"
by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
have I_Suc[simp]: "\<And>n. I (Suc n) = (if ?a n \<le> ?b n then I n + 1 else I n)"
by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
have J_Suc[simp]: "\<And>n. J (Suc n) = (if ?b n \<le> ?a n then J n + 1 else J n)"
by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
{ fix N have "\<And>p. z N p \<noteq> 0 \<Longrightarrow> \<exists>n<N. p = (I n, J n)"
by (induct N) (auto simp add: less_Suc_eq split: split_if_asm) }
{ fix i n assume "i < I n"
then have "(\<Sum>j. z n (i, j)) = x i"
oops
*)
end