mathlib documentation

probability.probability_mass_function.monad

Monad Operations for Probability Mass Functions #

This file constructs two operations on pmf that give it a monad structure. pure a is the distribution where a single value a has probability 1. bind pa pb : pmf β is the distribution given by sampling a : α from pa : pmf α, and then sampling from pb a : pmf β to get a final result b : β.

bind_on_support generalizes bind to allow binding to a partial function, so that the second argument only needs to be defined on the support of the first argument.

noncomputable def pmf.pure {α : Type u_1} (a : α) :
pmf α

The pure pmf is the pmf where all the mass lies in one point. The value of pure a is 1 at a and 0 elsewhere.

Equations
@[simp]
theorem pmf.pure_apply {α : Type u_1} (a a' : α) :
(pmf.pure a) a' = ite (a' = a) 1 0
@[simp]
theorem pmf.support_pure {α : Type u_1} (a : α) :
theorem pmf.mem_support_pure_iff {α : Type u_1} (a a' : α) :
a' (pmf.pure a).support a' = a
@[protected, instance]
noncomputable def pmf.inhabited {α : Type u_1} [inhabited α] :
Equations
@[simp]
theorem pmf.to_outer_measure_pure_apply {α : Type u_1} (a : α) (s : set α) :
@[simp]
theorem pmf.to_measure_pure_apply {α : Type u_1} (a : α) (s : set α) [measurable_space α] (hs : measurable_set s) :
((pmf.pure a).to_measure) s = ite (a s) 1 0

The measure of a set under pure a is 1 for sets containing a and 0 otherwise

@[protected]
theorem pmf.bind.summable {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (b : β) :
summable (λ (a : α), p a * (f a) b)
noncomputable def pmf.bind {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) :
pmf β

The monadic bind operation for pmf.

Equations
@[simp]
theorem pmf.bind_apply {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (b : β) :
(p.bind f) b = ∑' (a : α), p a * (f a) b
@[simp]
theorem pmf.support_bind {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) :
(p.bind f).support = {b : β | ∃ (a : α) (H : a p.support), b (f a).support}
theorem pmf.mem_support_bind_iff {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (b : β) :
b (p.bind f).support ∃ (a : α) (H : a p.support), b (f a).support
theorem pmf.coe_bind_apply {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (b : β) :
((p.bind f) b) = ∑' (a : α), (p a) * ((f a) b)
@[simp]
theorem pmf.pure_bind {α : Type u_1} {β : Type u_2} (a : α) (f : α → pmf β) :
(pmf.pure a).bind f = f a
@[simp]
theorem pmf.bind_pure {α : Type u_1} (p : pmf α) :
@[simp]
theorem pmf.bind_bind {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : pmf α) (f : α → pmf β) (g : β → pmf γ) :
(p.bind f).bind g = p.bind (λ (a : α), (f a).bind g)
theorem pmf.bind_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : pmf α) (q : pmf β) (f : α → β → pmf γ) :
p.bind (λ (a : α), q.bind (f a)) = q.bind (λ (b : β), p.bind (λ (a : α), f a b))
@[simp]
theorem pmf.to_outer_measure_bind_apply {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (s : set β) :
((p.bind f).to_outer_measure) s = ∑' (a : α), (p a) * ((f a).to_outer_measure) s
@[simp]
theorem pmf.to_measure_bind_apply {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) (s : set β) [measurable_space β] (hs : measurable_set s) :
((p.bind f).to_measure) s = ∑' (a : α), (p a) * ((f a).to_measure) s

The measure of a set under p.bind f is the sum over a : α of the probability of a under p times the measure of the set under f a

@[protected, instance]
noncomputable def pmf.monad  :
Equations
@[protected]
theorem pmf.bind_on_support.summable {α : Type u_1} {β : Type u_2} (p : pmf α) (f : Π (a : α), a p.supportpmf β) (b : β) :
summable (λ (a : α), p a * dite (p a = 0) (λ (h : p a = 0), 0) (λ (h : ¬p a = 0), (f a h) b))
noncomputable def pmf.bind_on_support {α : Type u_1} {β : Type u_2} (p : pmf α) (f : Π (a : α), a p.supportpmf β) :
pmf β

Generalized version of bind allowing f to only be defined on the support of p. p.bind f is equivalent to p.bind_on_support (λ a _, f a), see bind_on_support_eq_bind

Equations
@[simp]
theorem pmf.bind_on_support_apply {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (b : β) :
(p.bind_on_support f) b = ∑' (a : α), p a * dite (p a = 0) (λ (h : p a = 0), 0) (λ (h : ¬p a = 0), (f a h) b)
@[simp]
theorem pmf.support_bind_on_support {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) :
(p.bind_on_support f).support = {b : β | ∃ (a : α) (h : a p.support), b (f a h).support}
theorem pmf.mem_support_bind_on_support_iff {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (b : β) :
b (p.bind_on_support f).support ∃ (a : α) (h : a p.support), b (f a h).support
@[simp]
theorem pmf.bind_on_support_eq_bind {α : Type u_1} {β : Type u_2} (p : pmf α) (f : α → pmf β) :
p.bind_on_support (λ (a : α) (_x : a p.support), f a) = p.bind f

bind_on_support reduces to bind if f doesn't depend on the additional hypothesis

theorem pmf.coe_bind_on_support_apply {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (b : β) :
((p.bind_on_support f) b) = ∑' (a : α), (p a) * dite (p a = 0) (λ (h : p a = 0), 0) (λ (h : ¬p a = 0), ((f a h) b))
theorem pmf.bind_on_support_eq_zero_iff {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (b : β) :
(p.bind_on_support f) b = 0 ∀ (a : α) (ha : p a 0), (f a ha) b = 0
@[simp]
theorem pmf.pure_bind_on_support {α : Type u_1} {β : Type u_2} (a : α) (f : Π (a' : α), a' (pmf.pure a).supportpmf β) :
theorem pmf.bind_on_support_pure {α : Type u_1} (p : pmf α) :
p.bind_on_support (λ (a : α) (_x : a p.support), pmf.pure a) = p
@[simp]
theorem pmf.bind_on_support_bind_on_support {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : pmf α) (f : Π (a : α), a p.supportpmf β) (g : Π (b : β), b (p.bind_on_support f).supportpmf γ) :
(p.bind_on_support f).bind_on_support g = p.bind_on_support (λ (a : α) (ha : a p.support), (f a ha).bind_on_support (λ (b : β) (hb : b (f a ha).support), g b _))
theorem pmf.bind_on_support_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : pmf α) (q : pmf β) (f : Π (a : α), a p.supportΠ (b : β), b q.supportpmf γ) :
p.bind_on_support (λ (a : α) (ha : a p.support), q.bind_on_support (f a ha)) = q.bind_on_support (λ (b : β) (hb : b q.support), p.bind_on_support (λ (a : α) (ha : a p.support), f a ha b hb))
@[simp]
theorem pmf.to_outer_measure_bind_on_support_apply {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (s : set β) :
((p.bind_on_support f).to_outer_measure) s = ∑' (a : α), (p a) * dite (p a = 0) (λ (h : p a = 0), 0) (λ (h : ¬p a = 0), ((f a h).to_outer_measure) s)
@[simp]
theorem pmf.to_measure_bind_on_support_apply {α : Type u_1} {β : Type u_2} {p : pmf α} (f : Π (a : α), a p.supportpmf β) (s : set β) [measurable_space β] (hs : measurable_set s) :
((p.bind_on_support f).to_measure) s = ∑' (a : α), (p a) * dite (p a = 0) (λ (h : p a = 0), 0) (λ (h : ¬p a = 0), ((f a h).to_measure) s)

The measure of a set under p.bind_on_support f is the sum over a : α of the probability of a under p times the measure of the set under f a _. The additional if statement is needed since f is only a partial function