(* File: Pi_pmf.thy Authors: Manuel Eberl, Max W. Haslbeck *) section ‹Indexed products of PMFs› theory Pi_pmf imports "HOL-Probability.Probability" begin text ‹Conflicting notation from \<^theory>‹HOL-Analysis.Infinite_Sum›› no_notation Infinite_Sum.abs_summable_on (infixr "abs'_summable'_on" 46) subsection ‹Definition› text ‹ In analogy to @{const PiM}, we define an indexed product of PMFs. In the literature, this is typically called taking a vector of independent random variables. Note that the components do not have to be identically distributed. The operation takes an explicit index set \<^term>‹A :: 'a set› and a function \<^term>‹f :: 'a ⇒ 'b pmf› that maps each element from \<^term>‹A› to a PMF and defines the product measure $\bigotimes_{i\in A} f(i)$ , which is represented as a \<^typ>‹('a ⇒ 'b) pmf›. Note that unlike @{const PiM}, this only works for ∗‹finite› index sets. It could be extended to countable sets and beyond, but the construction becomes somewhat more involved. › definition Pi_pmf :: "'a set ⇒ 'b ⇒ ('a ⇒ 'b pmf) ⇒ ('a ⇒ 'b) pmf" where "Pi_pmf A dflt p = embed_pmf (λf. if (∀x. x ∉ A ⟶ f x = dflt) then ∏x∈A. pmf (p x) (f x) else 0)" text ‹ A technical subtlety that needs to be addressed is this: Intuitively, the functions in the support of a product distribution have domain ‹A›. However, since HOL is a total logic, these functions must still return ∗‹some› value for inputs outside ‹A›. The product measure @{const PiM} simply lets these functions return @{const undefined} in these cases. We chose a different solution here, which is to supply a default value \<^term>‹dflt :: 'b› that is returned in these cases. As one possible application, one could model the result of ‹n› different independent coin tosses as @{term "Pi_pmf {0..<n} False (λ_. bernoulli_pmf (1 / 2))"}. This returns a function of type \<^typ>‹nat ⇒ bool› that maps every natural number below ‹n› to the result of the corresponding coin toss, and every other natural number to \<^term>‹False›. › lemma pmf_Pi: assumes A: "finite A" shows "pmf (Pi_pmf A dflt p) f = (if (∀x. x ∉ A ⟶ f x = dflt) then ∏x∈A. pmf (p x) (f x) else 0)" unfolding Pi_pmf_def proof (rule pmf_embed_pmf, goal_cases) case 2 define S where "S = {f. ∀x. x ∉ A ⟶ f x = dflt}" define B where "B = (λx. set_pmf (p x))" have neutral_left: "(∏xa∈A. pmf (p xa) (f xa)) = 0" if "f ∈ PiE A B - (λf. restrict f A) ` S" for f proof - have "restrict (λx. if x ∈ A then f x else dflt) A ∈ (λf. restrict f A) ` S" by (intro imageI) (auto simp: S_def) also have "restrict (λx. if x ∈ A then f x else dflt) A = f" using that by (auto simp: PiE_def Pi_def extensional_def fun_eq_iff) finally show ?thesis using that by blast qed have neutral_right: "(∏xa∈A. pmf (p xa) (f xa)) = 0" if "f ∈ (λf. restrict f A) ` S - PiE A B" for f proof - from that obtain f' where f': "f = restrict f' A" "f' ∈ S" by auto moreover from this and that have "restrict f' A ∉ PiE A B" by simp then obtain x where "x ∈ A" "pmf (p x) (f' x) = 0" by (auto simp: B_def set_pmf_eq) with f' and A show ?thesis by auto qed have "(λf. ∏x∈A. pmf (p x) (f x)) abs_summable_on PiE A B" by (intro abs_summable_on_prod_PiE A) (auto simp: B_def) also have "?this ⟷ (λf. ∏x∈A. pmf (p x) (f x)) abs_summable_on (λf. restrict f A) ` S" by (intro abs_summable_on_cong_neutral neutral_left neutral_right) auto also have "… ⟷ (λf. ∏x∈A. pmf (p x) (restrict f A x)) abs_summable_on S" by (rule abs_summable_on_reindex_iff [symmetric]) (force simp: inj_on_def fun_eq_iff S_def) also have "… ⟷ (λf. if ∀x. x ∉ A ⟶ f x = dflt then ∏x∈A. pmf (p x) (f x) else 0) abs_summable_on UNIV" by (intro abs_summable_on_cong_neutral) (auto simp: S_def) finally have summable: … . have "1 = (∏x∈A. 1::real)" by simp also have "(∏x∈A. 1) = (∏x∈A. ∑⇩_{a}y∈B x. pmf (p x) y)" unfolding B_def by (subst infsetsum_pmf_eq_1) auto also have "(∏x∈A. ∑⇩_{a}y∈B x. pmf (p x) y) = (∑⇩_{a}f∈Pi⇩_{E}A B. ∏x∈A. pmf (p x) (f x))" by (intro infsetsum_prod_PiE [symmetric] A) (auto simp: B_def) also have "… = (∑⇩_{a}f∈(λf. restrict f A) ` S. ∏x∈A. pmf (p x) (f x))" using A by (intro infsetsum_cong_neutral neutral_left neutral_right refl) also have "… = (∑⇩_{a}f∈S. ∏x∈A. pmf (p x) (restrict f A x))" by (rule infsetsum_reindex) (force simp: inj_on_def fun_eq_iff S_def) also have "… = (∑⇩_{a}f∈S. ∏x∈A. pmf (p x) (f x))" by (intro infsetsum_cong) (auto simp: S_def) also have "… = (∑⇩_{a}f. if ∀x. x ∉ A ⟶ f x = dflt then ∏x∈A. pmf (p x) (f x) else 0)" by (intro infsetsum_cong_neutral) (auto simp: S_def) also have "ennreal … = (∫⇧^{+}f. ennreal (if ∀x. x ∉ A ⟶ f x = dflt then ∏x∈A. pmf (p x) (f x) else 0) ∂count_space UNIV)" by (intro nn_integral_conv_infsetsum [symmetric] summable) (auto simp: prod_nonneg) finally show ?case by simp qed (auto simp: prod_nonneg) lemma pmf_Pi': assumes "finite A" "⋀x. x ∉ A ⟹ f x = dflt" shows "pmf (Pi_pmf A dflt p) f = (∏x∈A. pmf (p x) (f x))" using assms by (subst pmf_Pi) auto lemma pmf_Pi_outside: assumes "finite A" "∃x. x ∉ A ∧ f x ≠ dflt" shows "pmf (Pi_pmf A dflt p) f = 0" using assms by (subst pmf_Pi) auto lemma pmf_Pi_empty [simp]: "Pi_pmf {} dflt p = return_pmf (λ_. dflt)" by (intro pmf_eqI, subst pmf_Pi) (auto simp: indicator_def) lemma set_Pi_pmf_subset: "finite A ⟹ set_pmf (Pi_pmf A dflt p) ⊆ {f. ∀x. x ∉ A ⟶ f x = dflt}" by (auto simp: set_pmf_eq pmf_Pi) lemma Pi_pmf_cong [cong]: assumes "A = A'" "dflt = dflt'" "⋀x. x ∈ A ⟹ f x = f' x" shows "Pi_pmf A dflt f = Pi_pmf A' dflt' f'" proof - have "(λg. ∏x∈A. pmf (f x) (g x)) = (λg. ∏x∈A. pmf (f' x) (g x))" by (intro ext prod.cong) (auto simp: assms) with assms show ?thesis by (simp add: Pi_pmf_def cong: if_cong) qed subsection ‹Dependent product sets with a default› text ‹ The following describes a dependent product of sets where the functions are required to return the default value \<^term>‹dflt› outside their domain, in analogy to @{const PiE}, which uses @{const undefined}. › definition PiE_dflt where "PiE_dflt A dflt B = {f. ∀x. (x ∈ A ⟶ f x ∈ B x) ∧ (x ∉ A ⟶ f x = dflt)}" lemma restrict_PiE_dflt: "(λh. restrict h A) ` PiE_dflt A dflt B = PiE A B" proof (intro equalityI subsetI) fix h assume "h ∈ (λh. restrict h A) ` PiE_dflt A dflt B" thus "h ∈ PiE A B" by (auto simp: PiE_dflt_def) next fix h assume h: "h ∈ PiE A B" hence "restrict (λx. if x ∈ A then h x else dflt) A ∈ (λh. restrict h A) ` PiE_dflt A dflt B" by (intro imageI) (auto simp: PiE_def extensional_def PiE_dflt_def) also have "restrict (λx. if x ∈ A then h x else dflt) A = h" using h by (auto simp: fun_eq_iff) finally show "h ∈ (λh. restrict h A) ` PiE_dflt A dflt B" . qed lemma dflt_image_PiE: "(λh x. if x ∈ A then h x else dflt) ` PiE A B = PiE_dflt A dflt B" (is "?f ` ?X = ?Y") proof (intro equalityI subsetI) fix h assume "h ∈ ?f ` ?X" thus "h ∈ ?Y" by (auto simp: PiE_dflt_def PiE_def) next fix h assume h: "h ∈ ?Y" hence "?f (restrict h A) ∈ ?f ` ?X" by (intro imageI) (auto simp: PiE_def extensional_def PiE_dflt_def) also have "?f (restrict h A) = h" using h by (auto simp: fun_eq_iff PiE_dflt_def) finally show "h ∈ ?f ` ?X" . qed lemma finite_PiE_dflt [intro]: assumes "finite A" "⋀x. x ∈ A ⟹ finite (B x)" shows "finite (PiE_dflt A d B)" proof - have "PiE_dflt A d B = (λf x. if x ∈ A then f x else d) ` PiE A B" by (rule dflt_image_PiE [symmetric]) also have "finite …" by (intro finite_imageI finite_PiE assms) finally show ?thesis . qed lemma card_PiE_dflt: assumes "finite A" "⋀x. x ∈ A ⟹ finite (B x)" shows "card (PiE_dflt A d B) = (∏x∈A. card (B x))" proof - from assms have "(∏x∈A. card (B x)) = card (PiE A B)" by (intro card_PiE [symmetric]) auto also have "PiE A B = (λf. restrict f A) ` PiE_dflt A d B" by (rule restrict_PiE_dflt [symmetric]) also have "card … = card (PiE_dflt A d B)" by (intro card_image) (force simp: inj_on_def restrict_def fun_eq_iff PiE_dflt_def) finally show ?thesis .. qed lemma PiE_dflt_empty_iff [simp]: "PiE_dflt A dflt B = {} ⟷ (∃x∈A. B x = {})" by (simp add: dflt_image_PiE [symmetric] PiE_eq_empty_iff) text ‹ The probability of an independent combination of events is precisely the product of the probabilities of each individual event. › lemma measure_Pi_pmf_PiE_dflt: assumes [simp]: "finite A" shows "measure_pmf.prob (Pi_pmf A dflt p) (PiE_dflt A dflt B) = (∏x∈A. measure_pmf.prob (p x) (B x))" proof - define B' where "B' = (λx. B x ∩ set_pmf (p x))" have "measure_pmf.prob (Pi_pmf A dflt p) (PiE_dflt A dflt B) = (∑⇩_{a}h∈PiE_dflt A dflt B. pmf (Pi_pmf A dflt p) h)" by (rule measure_pmf_conv_infsetsum) also have "… = (∑⇩_{a}h∈PiE_dflt A dflt B. ∏x∈A. pmf (p x) (h x))" by (intro infsetsum_cong, subst pmf_Pi') (auto simp: PiE_dflt_def) also have "… = (∑⇩_{a}h∈(λh. restrict h A) ` PiE_dflt A dflt B. ∏x∈A. pmf (p x) (h x))" by (subst infsetsum_reindex) (force simp: inj_on_def PiE_dflt_def fun_eq_iff)+ also have "(λh. restrict h A) ` PiE_dflt A dflt B = PiE A B" by (rule restrict_PiE_dflt) also have "(∑⇩_{a}h∈PiE A B. ∏x∈A. pmf (p x) (h x)) = (∑⇩_{a}h∈PiE A B'. ∏x∈A. pmf (p x) (h x))" by (intro infsetsum_cong_neutral) (auto simp: B'_def set_pmf_eq) also have "(∑⇩_{a}h∈PiE A B'. ∏x∈A. pmf (p x) (h x)) = (∏x∈A. infsetsum (pmf (p x)) (B' x))" by (intro infsetsum_prod_PiE) (auto simp: B'_def) also have "… = (∏x∈A. infsetsum (pmf (p x)) (B x))" by (intro prod.cong infsetsum_cong_neutral) (auto simp: B'_def set_pmf_eq) also have "… = (∏x∈A. measure_pmf.prob (p x) (B x))" by (subst measure_pmf_conv_infsetsum) (rule refl) finally show ?thesis . qed lemma set_Pi_pmf_subset': assumes "finite A" shows "set_pmf (Pi_pmf A dflt p) ⊆ PiE_dflt A dflt (set_pmf ∘ p)" using assms by (auto simp: set_pmf_eq pmf_Pi PiE_dflt_def) lemma Pi_pmf_return_pmf [simp]: assumes "finite A" shows "Pi_pmf A dflt (λx. return_pmf (f x)) = return_pmf (λx. if x ∈ A then f x else dflt)" proof - have "set_pmf (Pi_pmf A dflt (λx. return_pmf (f x))) ⊆ PiE_dflt A dflt (set_pmf ∘ (λx. return_pmf (f x)))" by (intro set_Pi_pmf_subset' assms) also have "… ⊆ {λx. if x ∈ A then f x else dflt}" by (auto simp: PiE_dflt_def) finally show ?thesis by (simp add: set_pmf_subset_singleton) qed lemma Pi_pmf_return_pmf' [simp]: assumes "finite A" shows "Pi_pmf A dflt (λ_. return_pmf dflt) = return_pmf (λ_. dflt)" using assms by simp lemma measure_Pi_pmf_Pi: fixes t::nat assumes [simp]: "finite A" shows "measure_pmf.prob (Pi_pmf A dflt p) (Pi A B) = (∏x∈A. measure_pmf.prob (p x) (B x))" (is "?lhs = ?rhs") proof - have "?lhs = measure_pmf.prob (Pi_pmf A dflt p) (PiE_dflt A dflt B)" by (intro measure_prob_cong_0) (auto simp: PiE_dflt_def PiE_def intro!: pmf_Pi_outside)+ also have "… = ?rhs" using assms by (simp add: measure_Pi_pmf_PiE_dflt) finally show ?thesis by simp qed subsection ‹Common PMF operations on products› text ‹ @{const Pi_pmf} distributes over the `bind' operation in the Giry monad: › lemma Pi_pmf_bind: assumes "finite A" shows "Pi_pmf A d (λx. bind_pmf (p x) (q x)) = do {f ← Pi_pmf A d' p; Pi_pmf A d (λx. q x (f x))}" (is "?lhs = ?rhs") proof (rule pmf_eqI, goal_cases) case (1 f) show ?case proof (cases "∃x∈-A. f x ≠ d") case False define B where "B = (λx. set_pmf (p x))" have [simp]: "countable (B x)" for x by (auto simp: B_def) { fix x :: 'a have "(λa. pmf (p x) a * 1) abs_summable_on B x" by (simp add: pmf_abs_summable) moreover have "norm (pmf (p x) a * 1) ≥ norm (pmf (p x) a * pmf (q x a) (f x))" for a unfolding norm_mult by (intro mult_left_mono) (auto simp: pmf_le_1) ultimately have "(λa. pmf (p x) a * pmf (q x a) (f x)) abs_summable_on B x" by (rule abs_summable_on_comparison_test) } note summable = this have "pmf ?rhs f = (∑⇩_{a}g. pmf (Pi_pmf A d' p) g * (∏x∈A. pmf (q x (g x)) (f x)))" by (subst pmf_bind, subst pmf_Pi') (insert assms False, simp_all add: pmf_expectation_eq_infsetsum) also have "… = (∑⇩_{a}g∈PiE_dflt A d' B. pmf (Pi_pmf A d' p) g * (∏x∈A. pmf (q x (g x)) (f x)))" unfolding B_def using assms by (intro infsetsum_cong_neutral) (auto simp: pmf_Pi PiE_dflt_def set_pmf_eq) also have "… = (∑⇩_{a}g∈PiE_dflt A d' B. (∏x∈A. pmf (p x) (g x) * pmf (q x (g x)) (f x)))" using assms by (intro infsetsum_cong) (auto simp: pmf_Pi PiE_dflt_def prod.distrib) also have "… = (∑⇩_{a}g∈(λg. restrict g A) ` PiE_dflt A d' B. (∏x∈A. pmf (p x) (g x) * pmf (q x (g x)) (f x)))" by (subst infsetsum_reindex) (force simp: PiE_dflt_def inj_on_def fun_eq_iff)+ also have "(λg. restrict g A) ` PiE_dflt A d' B = PiE A B" by (rule restrict_PiE_dflt) also have "(∑⇩_{a}g∈…. (∏x∈A. pmf (p x) (g x) * pmf (q x (g x)) (f x))) = (∏x∈A. ∑⇩_{a}a∈B x. pmf (p x) a * pmf (q x a) (f x))" using assms summable by (subst infsetsum_prod_PiE) simp_all also have "… = (∏x∈A. ∑⇩_{a}a. pmf (p x) a * pmf (q x a) (f x))" by (intro prod.cong infsetsum_cong_neutral) (auto simp: B_def set_pmf_eq) also have "… = pmf ?lhs f" using False assms by (subst pmf_Pi') (simp_all add: pmf_bind pmf_expectation_eq_infsetsum) finally show ?thesis .. next case True have "pmf ?rhs f = measure_pmf.expectation (Pi_pmf A d' p) (λx. pmf (Pi_pmf A d (λxa. q xa (x xa))) f)" using assms by (simp add: pmf_bind) also have "… = measure_pmf.expectation (Pi_pmf A d' p) (λx. 0)" using assms True by (intro Bochner_Integration.integral_cong pmf_Pi_outside) auto also have "… = pmf ?lhs f" using assms True by (subst pmf_Pi_outside) auto finally show ?thesis .. qed qed text ‹ Analogously any componentwise mapping can be pulled outside the product: › lemma Pi_pmf_map: assumes [simp]: "finite A" and "f dflt = dflt'" shows "Pi_pmf A dflt' (λx. map_pmf f (g x)) = map_pmf (λh. f ∘ h) (Pi_pmf A dflt g)" proof - have "Pi_pmf A dflt' (λx. map_pmf f (g x)) = Pi_pmf A dflt' (λx. g x ⤜ (λx. return_pmf (f x)))" using assms by (simp add: map_pmf_def Pi_pmf_bind) also have "… = Pi_pmf A dflt g ⤜ (λh. return_pmf (λx. if x ∈ A then f (h x) else dflt'))" by (subst Pi_pmf_bind[where d' = dflt]) auto also have "… = map_pmf (λh. f ∘ h) (Pi_pmf A dflt g)" unfolding map_pmf_def using set_Pi_pmf_subset'[of A dflt g] by (intro bind_pmf_cong refl arg_cong[of _ _ return_pmf]) (auto dest: simp: fun_eq_iff PiE_dflt_def assms(2)) finally show ?thesis . qed text ‹ We can exchange the default value in a product of PMFs like this: › lemma Pi_pmf_default_swap: assumes "finite A" shows "map_pmf (λf x. if x ∈ A then f x else dflt') (Pi_pmf A dflt p) = Pi_pmf A dflt' p" (is "?lhs = ?rhs") proof (rule pmf_eqI, goal_cases) case (1 f) let ?B = "(λf x. if x ∈ A then f x else dflt') -` {f} ∩ PiE_dflt A dflt (λ_. UNIV)" show ?case proof (cases "∃x∈-A. f x ≠ dflt'") case False let ?f' = "λx. if x ∈ A then f x else dflt" from False have "pmf ?lhs f = measure_pmf.prob (Pi_pmf A dflt p) ?B" using assms unfolding pmf_map by (intro measure_prob_cong_0) (auto simp: PiE_dflt_def pmf_Pi_outside) also from False have "?B = {?f'}" by (auto simp: fun_eq_iff PiE_dflt_def) also have "measure_pmf.prob (Pi_pmf A dflt p) {?f'} = pmf (Pi_pmf A dflt p) ?f'" by (simp add: measure_pmf_single) also have "… = pmf ?rhs f" using False assms by (subst (1 2) pmf_Pi) auto finally show ?thesis . next case True have "pmf ?lhs f = measure_pmf.prob (Pi_pmf A dflt p) ?B" using assms unfolding pmf_map by (intro measure_prob_cong_0) (auto simp: PiE_dflt_def pmf_Pi_outside) also from True have "?B = {}" by auto also have "measure_pmf.prob (Pi_pmf A dflt p) … = 0" by simp also have "0 = pmf ?rhs f" using True assms by (intro pmf_Pi_outside [symmetric]) auto finally show ?thesis . qed qed text ‹ The following rule allows reindexing the product: › lemma Pi_pmf_bij_betw: assumes "finite A" "bij_betw h A B" "⋀x. x ∉ A ⟹ h x ∉ B" shows "Pi_pmf A dflt (λ_. f) = map_pmf (λg. g ∘ h) (Pi_pmf B dflt (λ_. f))" (is "?lhs = ?rhs") proof - have B: "finite B" using assms bij_betw_finite by auto have "pmf ?lhs g = pmf ?rhs g" for g proof (cases "∀a. a ∉ A ⟶ g a = dflt") case True define h' where "h' = the_inv_into A h" have h': "h' (h x) = x" if "x ∈ A" for x unfolding h'_def using that assms by (auto simp add: bij_betw_def the_inv_into_f_f) have h: "h (h' x) = x" if "x ∈ B" for x unfolding h'_def using that assms f_the_inv_into_f_bij_betw by fastforce have "pmf ?rhs g = measure_pmf.prob (Pi_pmf B dflt (λ_. f)) ((λg. g ∘ h) -` {g})" unfolding pmf_map by simp also have "… = measure_pmf.prob (Pi_pmf B dflt (λ_. f)) (((λg. g ∘ h) -` {g}) ∩ PiE_dflt B dflt (λ_. UNIV))" using B by (intro measure_prob_cong_0) (auto simp: PiE_dflt_def pmf_Pi_outside) also have "… = pmf (Pi_pmf B dflt (λ_. f)) (λx. if x ∈ B then g (h' x) else dflt)" proof - have "(if h x ∈ B then g (h' (h x)) else dflt) = g x" for x using h' assms True by (cases "x ∈ A") (auto simp add: bij_betwE) then have "(λg. g ∘ h) -` {g} ∩ PiE_dflt B dflt (λ_. UNIV) = {(λx. if x ∈ B then g (h' x) else dflt)}" using assms h' h True unfolding PiE_dflt_def by auto then show ?thesis by (simp add: measure_pmf_single) qed also have "… = pmf (Pi_pmf A dflt (λ_. f)) g" using B assms True h'_def by (auto simp add: pmf_Pi intro!: prod.reindex_bij_betw bij_betw_the_inv_into) finally show ?thesis by simp next case False have "pmf ?rhs g = infsetsum (pmf (Pi_pmf B dflt (λ_. f))) ((λg. g ∘ h) -` {g})" using assms by (auto simp add: measure_pmf_conv_infsetsum pmf_map) also have "… = infsetsum (λ_. 0) ((λg x. g (h x)) -` {g})" using B False assms by (intro infsetsum_cong pmf_Pi_outside) fastforce+ also have "… = 0" by simp finally show ?thesis using assms False by (auto simp add: pmf_Pi pmf_map) qed then show ?thesis by (rule pmf_eqI) qed text ‹ A product of uniform random choices is again a uniform distribution. › lemma Pi_pmf_of_set: assumes "finite A" "⋀x. x ∈ A ⟹ finite (B x)" "⋀x. x ∈ A ⟹ B x ≠ {}" shows "Pi_pmf A d (λx. pmf_of_set (B x)) = pmf_of_set (PiE_dflt A d B)" (is "?lhs = ?rhs") proof (rule pmf_eqI, goal_cases) case (1 f) show ?case proof (cases "∃x. x ∉ A ∧ f x ≠ d") case True hence "pmf ?lhs f = 0" using assms by (intro pmf_Pi_outside) (auto simp: PiE_dflt_def) also from True have "f ∉ PiE_dflt A d B" by (auto simp: PiE_dflt_def) hence "0 = pmf ?rhs f" using assms by (subst pmf_of_set) auto finally show ?thesis . next case False hence "pmf ?lhs f = (∏x∈A. pmf (pmf_of_set (B x)) (f x))" using assms by (subst pmf_Pi') auto also have "… = (∏x∈A. indicator (B x) (f x) / real (card (B x)))" by (intro prod.cong refl, subst pmf_of_set) (use assms False in auto) also have "… = (∏x∈A. indicator (B x) (f x)) / real (∏x∈A. card (B x))" by (subst prod_dividef) simp_all also have "(∏x∈A. indicator (B x) (f x) :: real) = indicator (PiE_dflt A d B) f" using assms False by (auto simp: indicator_def PiE_dflt_def) also have "(∏x∈A. card (B x)) = card (PiE_dflt A d B)" using assms by (intro card_PiE_dflt [symmetric]) auto also have "indicator (PiE_dflt A d B) f / … = pmf ?rhs f" using assms by (intro pmf_of_set [symmetric]) auto finally show ?thesis . qed qed subsection ‹Merging and splitting PMF products› text ‹ The following lemma shows that we can add a single PMF to a product: › lemma Pi_pmf_insert: assumes "finite A" "x ∉ A" shows "Pi_pmf (insert x A) dflt p = map_pmf (λ(y,f). f(x:=y)) (pair_pmf (p x) (Pi_pmf A dflt p))" proof (intro pmf_eqI) fix f let ?M = "pair_pmf (p x) (Pi_pmf A dflt p)" have "pmf (map_pmf (λ(y, f). f(x := y)) ?M) f = measure_pmf.prob ?M ((λ(y, f). f(x := y)) -` {f})" by (subst pmf_map) auto also have "((λ(y, f). f(x := y)) -` {f}) = (⋃y'. {(f x, f(x := y'))})" by (auto simp: fun_upd_def fun_eq_iff) also have "measure_pmf.prob ?M … = measure_pmf.prob ?M {(f x, f(x := dflt))}" using assms by (intro measure_prob_cong_0) (auto simp: pmf_pair pmf_Pi split: if_splits) also have "… = pmf (p x) (f x) * pmf (Pi_pmf A dflt p) (f(x := dflt))" by (simp add: measure_pmf_single pmf_pair pmf_Pi) also have "… = pmf (Pi_pmf (insert x A) dflt p) f" proof (cases "∀y. y ∉ insert x A ⟶ f y = dflt") case True with assms have "pmf (p x) (f x) * pmf (Pi_pmf A dflt p) (f(x := dflt)) = pmf (p x) (f x) * (∏xa∈A. pmf (p xa) ((f(x := dflt)) xa))" by (subst pmf_Pi') auto also have "(∏xa∈A. pmf (p xa) ((f(x := dflt)) xa)) = (∏xa∈A. pmf (p xa) (f xa))" using assms by (intro prod.cong) auto also have "pmf (p x) (f x) * … = pmf (Pi_pmf (insert x A) dflt p) f" using assms True by (subst pmf_Pi') auto finally show ?thesis . qed (insert assms, auto simp: pmf_Pi) finally show "… = pmf (map_pmf (λ(y, f). f(x := y)) ?M) f" .. qed lemma Pi_pmf_insert': assumes "finite A" "x ∉ A" shows "Pi_pmf (insert x A) dflt p = do {y ← p x; f ← Pi_pmf A dflt p; return_pmf (f(x := y))}" using assms by (subst Pi_pmf_insert) (auto simp add: map_pmf_def pair_pmf_def case_prod_beta' bind_return_pmf bind_assoc_pmf) lemma Pi_pmf_singleton: "Pi_pmf {x} dflt p = map_pmf (λa b. if b = x then a else dflt) (p x)" proof - have "Pi_pmf {x} dflt p = map_pmf (fun_upd (λ_. dflt) x) (p x)" by (subst Pi_pmf_insert) (simp_all add: pair_return_pmf2 pmf.map_comp o_def) also have "fun_upd (λ_. dflt) x = (λz y. if y = x then z else dflt)" by (simp add: fun_upd_def fun_eq_iff) finally show ?thesis . qed text ‹ Projecting a product of PMFs onto a component yields the expected result: › lemma Pi_pmf_component: assumes "finite A" shows "map_pmf (λf. f x) (Pi_pmf A dflt p) = (if x ∈ A then p x else return_pmf dflt)" proof (cases "x ∈ A") case True define A' where "A' = A - {x}" from assms and True have A': "A = insert x A'" by (auto simp: A'_def) from assms have "map_pmf (λf. f x) (Pi_pmf A dflt p) = p x" unfolding A' by (subst Pi_pmf_insert) (auto simp: A'_def pmf.map_comp o_def case_prod_unfold map_fst_pair_pmf) with True show ?thesis by simp next case False have "map_pmf (λf. f x) (Pi_pmf A dflt p) = map_pmf (λ_. dflt) (Pi_pmf A dflt p)" using assms False set_Pi_pmf_subset[of A dflt p] by (intro pmf.map_cong refl) (auto simp: set_pmf_eq pmf_Pi_outside) with False show ?thesis by simp qed text ‹ We can take merge two PMF products on disjoint sets like this: › lemma Pi_pmf_union: assumes "finite A" "finite B" "A ∩ B = {}" shows "Pi_pmf (A ∪ B) dflt p = map_pmf (λ(f,g) x. if x ∈ A then f x else g x) (pair_pmf (Pi_pmf A dflt p) (Pi_pmf B dflt p))" (is "_ = map_pmf (?h A) (?q A)") using assms(1,3) proof (induction rule: finite_induct) case (insert x A) have "map_pmf (?h (insert x A)) (?q (insert x A)) = do {v ← p x; (f, g) ← pair_pmf (Pi_pmf A dflt p) (Pi_pmf B dflt p); return_pmf (λy. if y ∈ insert x A then (f(x := v)) y else g y)}" by (subst Pi_pmf_insert) (insert insert.hyps insert.prems, simp_all add: pair_pmf_def map_bind_pmf bind_map_pmf bind_assoc_pmf bind_return_pmf) also have "… = do {v ← p x; (f, g) ← ?q A; return_pmf ((?h A (f,g))(x := v))}" by (intro bind_pmf_cong refl) (auto simp: fun_eq_iff) also have "… = do {v ← p x; f ← map_pmf (?h A) (?q A); return_pmf (f(x := v))}" by (simp add: bind_map_pmf map_bind_pmf case_prod_unfold cong: if_cong) also have "… = do {v ← p x; f ← Pi_pmf (A ∪ B) dflt p; return_pmf (f(x := v))}" using insert.hyps and insert.prems by (intro bind_pmf_cong insert.IH [symmetric] refl) auto also have "… = Pi_pmf (insert x (A ∪ B)) dflt p" by (subst Pi_pmf_insert) (insert assms insert.hyps insert.prems, auto simp: pair_pmf_def map_bind_pmf) also have "insert x (A ∪ B) = insert x A ∪ B" by simp finally show ?case .. qed (simp_all add: case_prod_unfold map_snd_pair_pmf) text ‹ We can also project a product to a subset of the indices by mapping all the other indices to the default value: › lemma Pi_pmf_subset: assumes "finite A" "A' ⊆ A" shows "Pi_pmf A' dflt p = map_pmf (λf x. if x ∈ A' then f x else dflt) (Pi_pmf A dflt p)" proof - let ?P = "pair_pmf (Pi_pmf A' dflt p) (Pi_pmf (A - A') dflt p)" from assms have [simp]: "finite A'" by (blast dest: finite_subset) from assms have "A = A' ∪ (A - A')" by blast also have "Pi_pmf … dflt p = map_pmf (λ(f,g) x. if x ∈ A' then f x else g x) ?P" using assms by (intro Pi_pmf_union) auto also have "map_pmf (λf x. if x ∈ A' then f x else dflt) … = map_pmf fst ?P" unfolding map_pmf_comp o_def case_prod_unfold using set_Pi_pmf_subset[of A' dflt p] by (intro map_pmf_cong refl) (auto simp: fun_eq_iff) also have "… = Pi_pmf A' dflt p" by (simp add: map_fst_pair_pmf) finally show ?thesis .. qed lemma Pi_pmf_subset': fixes f :: "'a ⇒ 'b pmf" assumes "finite A" "B ⊆ A" "⋀x. x ∈ A - B ⟹ f x = return_pmf dflt" shows "Pi_pmf A dflt f = Pi_pmf B dflt f" proof - have "Pi_pmf (B ∪ (A - B)) dflt f = map_pmf (λ(f, g) x. if x ∈ B then f x else g x) (pair_pmf (Pi_pmf B dflt f) (Pi_pmf (A - B) dflt f))" using assms by (intro Pi_pmf_union) (auto dest: finite_subset) also have "Pi_pmf (A - B) dflt f = Pi_pmf (A - B) dflt (λ_. return_pmf dflt)" using assms by (intro Pi_pmf_cong) auto also have "… = return_pmf (λ_. dflt)" using assms by simp also have "map_pmf (λ(f, g) x. if x ∈ B then f x else g x) (pair_pmf (Pi_pmf B dflt f) (return_pmf (λ_. dflt))) = map_pmf (λf x. if x ∈ B then f x else dflt) (Pi_pmf B dflt f)" by (simp add: map_pmf_def pair_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') also have "… = Pi_pmf B dflt f" using assms by (intro Pi_pmf_default_swap) (auto dest: finite_subset) also have "B ∪ (A - B) = A" using assms by auto finally show ?thesis . qed lemma Pi_pmf_if_set: assumes "finite A" shows "Pi_pmf A dflt (λx. if b x then f x else return_pmf dflt) = Pi_pmf {x∈A. b x} dflt f" proof - have "Pi_pmf A dflt (λx. if b x then f x else return_pmf dflt) = Pi_pmf {x∈A. b x} dflt (λx. if b x then f x else return_pmf dflt)" using assms by (intro Pi_pmf_subset') auto also have "… = Pi_pmf {x∈A. b x} dflt f" by (intro Pi_pmf_cong) auto finally show ?thesis . qed lemma Pi_pmf_if_set': assumes "finite A" shows "Pi_pmf A dflt (λx. if b x then return_pmf dflt else f x) = Pi_pmf {x∈A. ¬b x} dflt f" proof - have "Pi_pmf A dflt (λx. if b x then return_pmf dflt else f x) = Pi_pmf {x∈A. ¬b x} dflt (λx. if b x then return_pmf dflt else f x)" using assms by (intro Pi_pmf_subset') auto also have "… = Pi_pmf {x∈A. ¬b x} dflt f" by (intro Pi_pmf_cong) auto finally show ?thesis . qed text ‹ Lastly, we can delete a single component from a product: › lemma Pi_pmf_remove: assumes "finite A" shows "Pi_pmf (A - {x}) dflt p = map_pmf (λf. f(x := dflt)) (Pi_pmf A dflt p)" proof - have "Pi_pmf (A - {x}) dflt p = map_pmf (λf xa. if xa ∈ A - {x} then f xa else dflt) (Pi_pmf A dflt p)" using assms by (intro Pi_pmf_subset) auto also have "… = map_pmf (λf. f(x := dflt)) (Pi_pmf A dflt p)" using set_Pi_pmf_subset[of A dflt p] assms by (intro map_pmf_cong refl) (auto simp: fun_eq_iff) finally show ?thesis . qed subsection ‹Applications› text ‹ Choosing a subset of a set uniformly at random is equivalent to tossing a fair coin independently for each element and collecting all the elements that came up heads. › lemma pmf_of_set_Pow_conv_bernoulli: assumes "finite (A :: 'a set)" shows "map_pmf (λb. {x∈A. b x}) (Pi_pmf A P (λ_. bernoulli_pmf (1/2))) = pmf_of_set (Pow A)" proof - have "Pi_pmf A P (λ_. bernoulli_pmf (1/2)) = pmf_of_set (PiE_dflt A P (λx. UNIV))" using assms by (simp add: bernoulli_pmf_half_conv_pmf_of_set Pi_pmf_of_set) also have "map_pmf (λb. {x∈A. b x}) … = pmf_of_set (Pow A)" proof - have "bij_betw (λb. {x ∈ A. b x}) (PiE_dflt A P (λ_. UNIV)) (Pow A)" by (rule bij_betwI[of _ _ _ "λB b. if b ∈ A then b ∈ B else P"]) (auto simp add: PiE_dflt_def) then show ?thesis using assms by (intro map_pmf_of_set_bij_betw) auto qed finally show ?thesis by simp qed text ‹ A binomial distribution can be seen as the number of successes in ‹n› independent coin tosses. › lemma binomial_pmf_altdef': fixes A :: "'a set" assumes "finite A" and "card A = n" and p: "p ∈ {0..1}" shows "binomial_pmf n p = map_pmf (λf. card {x∈A. f x}) (Pi_pmf A dflt (λ_. bernoulli_pmf p))" (is "?lhs = ?rhs") proof - from assms have "?lhs = binomial_pmf (card A) p" by simp also have "… = ?rhs" using assms(1) proof (induction rule: finite_induct) case empty with p show ?case by (simp add: binomial_pmf_0) next case (insert x A) from insert.hyps have "card (insert x A) = Suc (card A)" by simp also have "binomial_pmf … p = do { b ← bernoulli_pmf p; f ← Pi_pmf A dflt (λ_. bernoulli_pmf p); return_pmf ((if b then 1 else 0) + card {y ∈ A. f y}) }" using p by (simp add: binomial_pmf_Suc insert.IH bind_map_pmf) also have "… = do { b ← bernoulli_pmf p; f ← Pi_pmf A dflt (λ_. bernoulli_pmf p); return_pmf (card {y ∈ insert x A. (f(x := b)) y}) }" proof (intro bind_pmf_cong refl, goal_cases) case (1 b f) have "(if b then 1 else 0) + card {y∈A. f y} = card ((if b then {x} else {}) ∪ {y∈A. f y})" using insert.hyps by auto also have "(if b then {x} else {}) ∪ {y∈A. f y} = {y∈insert x A. (f(x := b)) y}" using insert.hyps by auto finally show ?case by simp qed also have "… = map_pmf (λf. card {y∈insert x A. f y}) (Pi_pmf (insert x A) dflt (λ_. bernoulli_pmf p))" using insert.hyps by (subst Pi_pmf_insert) (simp_all add: pair_pmf_def map_bind_pmf) finally show ?case . qed finally show ?thesis . qed end