author paulson Tue, 10 Feb 2015 16:09:30 +0000 changeset 59505 d64d48eb71cc parent 59504 8c6747dba731 (current diff) parent 59496 6faf024a1893 (diff) child 59506 4af607652318
Merge
```--- a/src/HOL/Probability/Probability_Mass_Function.thy	Tue Feb 10 16:08:11 2015 +0000
+++ b/src/HOL/Probability/Probability_Mass_Function.thy	Tue Feb 10 16:09:30 2015 +0000
@@ -12,6 +12,9 @@
"~~/src/HOL/Library/Multiset"
begin

+lemma ereal_divide': "b \<noteq> 0 \<Longrightarrow> ereal (a / b) = ereal a / ereal b"
+  using ereal_divide[of a b] by simp
+
lemma (in finite_measure) countable_support:
"countable {x. measure M {x} \<noteq> 0}"
proof cases
@@ -688,6 +691,11 @@
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)

+lemma ereal_pmf_join: "ereal (pmf (join_pmf N) i) = (\<integral>\<^sup>+M. pmf M i \<partial>measure_pmf N)"
+  unfolding pmf_join
+  by (intro nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1])
+     (auto simp: pmf_le_1 pmf_nonneg)
+
lemma set_pmf_join_pmf: "set_pmf (join_pmf f) = (\<Union>p\<in>set_pmf f. set_pmf p)"
apply(subst integral_nonneg_eq_0_iff_AE)
@@ -735,6 +743,9 @@
unfolding f by (subst bind_distr[OF _ measurable_measure_pmf]) auto
qed

+lemma ereal_pmf_bind: "pmf (bind_pmf N f) i = (\<integral>\<^sup>+x. pmf (f x) i \<partial>measure_pmf N)"
+  by (auto intro!: nn_integral_distr simp: bind_pmf_def ereal_pmf_join map_pmf.rep_eq)
+
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)

@@ -857,6 +868,12 @@

end

+lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (\<lambda>x. map_pmf f (g x))"
+  unfolding bind_return_pmf''[symmetric] bind_assoc_pmf[of M] ..
+
+lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
+  unfolding bind_return_pmf''[symmetric] bind_assoc_pmf bind_return_pmf ..
+
lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)"
unfolding bind_pmf_def[symmetric]
unfolding bind_return_pmf''[symmetric] join_eq_bind_pmf bind_assoc_pmf
@@ -982,6 +999,94 @@
by (auto simp: pmf.rep_eq map_pmf.rep_eq measure_distr AE_measure_pmf_iff inj_onD
intro!: measure_pmf.finite_measure_eq_AE)

+subsection \<open> Conditional Probabilities \<close>
+
+context
+  fixes p :: "'a pmf" and s :: "'a set"
+  assumes not_empty: "set_pmf p \<inter> s \<noteq> {}"
+begin
+
+interpretation pmf_as_measure .
+
+lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s \<noteq> 0"
+proof
+  assume "emeasure (measure_pmf p) s = 0"
+  then have "AE x in measure_pmf p. x \<notin> s"
+    by (rule AE_I[rotated]) auto
+  with not_empty show False
+    by (auto simp: AE_measure_pmf_iff)
+qed
+
+lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s \<noteq> 0"
+  using emeasure_measure_pmf_not_zero unfolding measure_pmf.emeasure_eq_measure by simp
+
+lift_definition cond_pmf :: "'a pmf" is
+  "uniform_measure (measure_pmf p) s"
+proof (intro conjI)
+  show "prob_space (uniform_measure (measure_pmf p) s)"
+    by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero)
+  show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} \<noteq> 0"
+    by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure
+                  AE_measure_pmf_iff set_pmf.rep_eq)
+qed simp
+
+lemma pmf_cond: "pmf cond_pmf x = (if x \<in> s then pmf p x / measure p s else 0)"
+  by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq)
+
+lemma set_cond_pmf: "set_pmf cond_pmf = set_pmf p \<inter> s"
+  by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: split_if_asm)
+
+end
+
+lemma cond_map_pmf:
+  assumes "set_pmf p \<inter> f -` s \<noteq> {}"
+  shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
+proof -
+  have *: "set_pmf (map_pmf f p) \<inter> s \<noteq> {}"
+    using assms by (simp add: set_map_pmf) auto
+  { fix x
+    have "ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x) =
+      emeasure p (f -` s \<inter> f -` {x}) / emeasure p (f -` s)"
+      unfolding ereal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure)
+    also have "f -` s \<inter> f -` {x} = (if x \<in> s then f -` {x} else {})"
+      by auto
+    also have "emeasure p (if x \<in> s then f -` {x} else {}) / emeasure p (f -` s) =
+      ereal (pmf (cond_pmf (map_pmf f p) s) x)"
+      using measure_measure_pmf_not_zero[OF *]
+      by (simp add: pmf_cond[OF *] ereal_divide' ereal_pmf_map measure_pmf.emeasure_eq_measure[symmetric]
+               del: ereal_divide)
+    finally have "ereal (pmf (cond_pmf (map_pmf f p) s) x) = ereal (pmf (map_pmf f (cond_pmf p (f -` s))) x)"
+      by simp }
+  then show ?thesis
+    by (intro pmf_eqI) simp
+qed
+
+lemma bind_cond_pmf_cancel:
+  assumes in_S: "\<And>x. x \<in> set_pmf p \<Longrightarrow> x \<in> S x"
+  assumes S_eq: "\<And>x y. x \<in> S y \<Longrightarrow> S x = S y"
+  shows "bind_pmf p (\<lambda>x. cond_pmf p (S x)) = p"
+proof (rule pmf_eqI)
+  have [simp]: "\<And>x. x \<in> p \<Longrightarrow> p \<inter> (S x) \<noteq> {}"
+    using in_S by auto
+  fix z
+  have pmf_le: "pmf p z \<le> measure p (S z)"
+  proof cases
+    assume "z \<in> p" from in_S[OF this] show ?thesis
+      by (auto intro!: measure_pmf.finite_measure_mono simp: pmf.rep_eq)
+  qed (simp add: set_pmf_iff measure_nonneg)
+
+  have "ereal (pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z) =
+    (\<integral>\<^sup>+ x. ereal (pmf p z / measure p (S z)) * indicator (S z) x \<partial>p)"
+    by (subst ereal_pmf_bind)
+       (auto intro!: nn_integral_cong_AE dest!: S_eq split: split_indicator
+             simp: AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf in_S)
+  also have "\<dots> = pmf p z"
+    using pmf_le pmf_nonneg[of p z]
+    by (subst nn_integral_cmult) (simp_all add: measure_nonneg measure_pmf.emeasure_eq_measure)
+  finally show "pmf (bind_pmf p (\<lambda>x. cond_pmf p (S x))) z = pmf p z"
+    by simp
+qed
+
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
for R p q
where
@@ -1026,89 +1131,23 @@
from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto

-    note pmf_nonneg[intro, simp]
-    let ?pq = "\<lambda>y x. pmf pq (x, y)"
-    let ?qr = "\<lambda>y z. pmf qr (y, z)"
-
-    have nn_integral_pp2: "\<And>y. (\<integral>\<^sup>+ x. ?pq y x \<partial>count_space UNIV) = pmf q y"
-      by (simp add: nn_integral_pmf' inj_on_def q)
-         (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
-    have nn_integral_rr1: "\<And>y. (\<integral>\<^sup>+ x. ?qr y x \<partial>count_space UNIV) = pmf q y"
-      by (simp add: nn_integral_pmf' inj_on_def q')
-         (auto simp add: ereal_pmf_map intro!: arg_cong2[where f=emeasure])
-    have eq: "\<And>y. (\<integral>\<^sup>+ x. ?pq y x \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?qr y z \<partial>count_space UNIV)"
-
-    def assign \<equiv> "\<lambda>(z, x, y). ?pq y x * ?qr y z / pmf q y"
-    have assign_nonneg [simp]: "\<And>z x y. 0 \<le> assign (z, x, y)" by(simp add: assign_def)
-    have assign_eq_0_outside: "\<And>z x y. \<lbrakk> ?pq y x = 0 \<or> ?qr y z = 0 \<rbrakk> \<Longrightarrow> assign (z, x, y) = 0"
-    have nn_integral_assign1: "\<And>z y. (\<integral>\<^sup>+ x. assign (z, x, y) \<partial>count_space UNIV) = ?qr y z"
-    proof -
-      fix y z
-      have "(\<integral>\<^sup>+ x. assign (z, x, y) \<partial>count_space UNIV) =
-            (\<integral>\<^sup>+ x. ?pq y x \<partial>count_space UNIV) * (?qr y z / pmf q y)"
-        by(simp add: assign_def nn_integral_multc times_ereal.simps(1)[symmetric] divide_real_def mult.assoc del: times_ereal.simps(1))
-      also have "\<dots> = ?qr y z" by(auto simp add: image_iff q' pmf_eq_0_set_pmf set_map_pmf nn_integral_pp2)
-      finally show "?thesis z y" .
-    qed
-    have nn_integral_assign2: "\<And>x y. (\<integral>\<^sup>+ z. assign (z, x, y) \<partial>count_space UNIV) = ?pq y x"
-    proof -
-      fix x y
-      have "(\<integral>\<^sup>+ z. assign (z, x, y) \<partial>count_space UNIV) = (\<integral>\<^sup>+ z. ?qr y z \<partial>count_space UNIV) * (?pq y x / pmf q y)"
-        by(simp add: assign_def divide_real_def mult.commute[where a="?pq y x"] mult.assoc nn_integral_multc times_ereal.simps(1)[symmetric] del: times_ereal.simps(1))
-      also have "\<dots> = ?pq y x" by(auto simp add: image_iff pmf_eq_0_set_pmf set_map_pmf q nn_integral_rr1)
-      finally show "?thesis x y" .
-    qed
-
-    def pqr \<equiv> "embed_pmf assign"
-    { fix z x y
-      have "assign (z, x, y) = pmf pqr (z, x, y)"
-        unfolding pqr_def
-      proof (subst pmf_embed_pmf)
-        have "(\<integral>\<^sup>+ zxy. ereal (assign zxy) \<partial>count_space UNIV) =
-              (\<integral>\<^sup>+ xy. \<integral>\<^sup>+ z. ereal (assign (z, xy)) \<partial>count_space UNIV \<partial>count_space UNIV)"
-          by(subst nn_integral_snd_count_space) simp
-        also have "\<dots> = (\<integral>\<^sup>+ z. ?pq (snd z) (fst z) \<partial>count_space UNIV)"
-          by (subst nn_integral_assign2[symmetric]) (auto intro!: nn_integral_cong)
-        finally show "(\<integral>\<^sup>+ zxy. ereal (assign zxy) \<partial>count_space UNIV) = 1"
-          by (simp add: nn_integral_pmf emeasure_pmf)
-      qed auto }
-    note a = this
-
-    def pr \<equiv> "map_pmf (\<lambda>(z, x, y). (x, z)) pqr"
+    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)))"
+    have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {(y', z). y' = y} \<noteq> {}"
+      by (force simp: q' set_map_pmf)

have "rel_pmf (R OO S) p r"
-    proof
-      have pq_eq: "pq = map_pmf (\<lambda>(z, x, y). (x, y)) pqr"
-      proof (rule pmf_eqI)
-        fix i
-        show "pmf pq i = pmf (map_pmf (\<lambda>(z, x, y). (x, y)) pqr) i"
-          using nn_integral_assign2[of "fst i" "snd i", symmetric]
-          by(cases i)
-            (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
-                  simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
-      qed
-      then show "map_pmf fst pr = p"
-        unfolding p pr_def by (simp add: map_pmf_comp split_beta)
-
-      have qr_eq: "qr = map_pmf (\<lambda>(z, x, y). (y, z)) pqr"
-      proof (rule pmf_eqI)
-        fix i show "pmf qr i = pmf (map_pmf (\<lambda>(z, x, y). (y, z)) pqr) i"
-          using nn_integral_assign1[of "snd i" "fst i", symmetric]
-          by(cases i)
-            (auto simp add: a nn_integral_pmf' inj_on_def ereal.inject[symmetric] ereal_pmf_map
-                  simp del: ereal.inject intro!: arg_cong2[where f=emeasure])
-      qed
-      then show "map_pmf snd pr = r"
-        unfolding r pr_def by (simp add: map_pmf_comp split_beta)
-
-      fix x z assume "(x, z) \<in> set_pmf pr"
-      then have "\<exists>y. (x, y) \<in> set_pmf pq \<and> (y, z) \<in> set_pmf qr"
-        unfolding pr_def pq_eq qr_eq by (force simp: set_map_pmf)
+    proof (rule rel_pmf.intros)
+      fix x z assume "(x, z) \<in> pr"
+      then have "\<exists>y. (x, y) \<in> pq \<and> (y, z) \<in> qr"
+        by (auto simp: q pr_welldefined pr_def set_bind_pmf split_beta set_return_pmf set_cond_pmf set_map_pmf)
with pq qr show "(R OO S) x z"
by blast
-    qed }
+    next
+      have "map_pmf snd pr = map_pmf snd (bind_pmf q (\<lambda>y. cond_pmf qr {(y', z). y' = y}))"
+        by (simp add: pr_def q split_beta bind_map_pmf bind_return_pmf'' map_bind_pmf map_return_pmf)
+      then show "map_pmf snd pr = r"
+        unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) auto
+    qed (simp add: pr_def map_bind_pmf split_beta map_return_pmf bind_return_pmf'' p) }
then show "rel_pmf R OO rel_pmf S \<le> rel_pmf (R OO S)"