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.

bindOnSupport 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.

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.

@[simp]

theorem Pmf.pure_apply {α : Type u_1} (a : α) (a' : α) : ↑(Pmf.pure a) a' = if a' = a then 1 else 0

@[simp]

theorem Pmf.support_pure {α : Type u_1} (a : α) : Pmf.support (Pmf.pure a) = {a}

theorem Pmf.mem_support_pure_iff {α : Type u_1} (a : α) (a' : α) : a' ∈ Pmf.support (Pmf.pure a) ↔ a' = a

theorem Pmf.pure_apply_self {α : Type u_1} (a : α) : ↑(Pmf.pure a) a = 1

theorem Pmf.pure_apply_of_ne {α : Type u_1} (a : α) (a' : α) (h : a' ≠ a) : ↑(Pmf.pure a) a' = 0

instance Pmf.instInhabitedPmf {α : Type u_1} [Inhabited α] : Inhabited (Pmf α)

@[simp]

theorem Pmf.toOuterMeasure_pure_apply {α : Type u_1} (a : α) (s : Set α) : ↑(Pmf.toOuterMeasure (Pmf.pure a)) s = if a ∈ s then 1 else 0

@[simp]

theorem Pmf.toMeasure_pure_apply {α : Type u_1} (a : α) (s : Set α) [MeasurableSpace α] (hs : MeasurableSet s) : ↑↑(Pmf.toMeasure (Pmf.pure a)) s = if a ∈ s then 1 else 0

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

theorem Pmf.toMeasure_pure {α : Type u_1} (a : α) [MeasurableSpace α] : Pmf.toMeasure (Pmf.pure a) = MeasureTheory.Measure.dirac a

@[simp]

theorem Pmf.toPmf_dirac {α : Type u_1} (a : α) [MeasurableSpace α] [Countable α] [h : MeasurableSingletonClass α] : MeasureTheory.Measure.toPmf (MeasureTheory.Measure.dirac a) = Pmf.pure a

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.

@[simp]

theorem Pmf.bind_apply {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : α → Pmf β) (b : β) : ↑(Pmf.bind p 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) : Pmf.support (Pmf.bind p f) = ⋃ (a : α) (_ : a ∈ Pmf.support p), Pmf.support (f a)

theorem Pmf.mem_support_bind_iff {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : α → Pmf β) (b : β) : b ∈ Pmf.support (Pmf.bind p f) ↔ ∃ a, a ∈ Pmf.support p ∧ b ∈ Pmf.support (f a)

@[simp]

theorem Pmf.pure_bind {α : Type u_1} {β : Type u_2} (a : α) (f : α → Pmf β) : Pmf.bind (Pmf.pure a) f = f a

@[simp]

theorem Pmf.bind_pure {α : Type u_1} (p : Pmf α) : Pmf.bind p Pmf.pure = p

@[simp]

theorem Pmf.bind_const {α : Type u_1} {β : Type u_2} (p : Pmf α) (q : Pmf β) : (Pmf.bind p fun x => q) = q

@[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) : Pmf.bind (Pmf.bind p f) g = Pmf.bind p fun a => Pmf.bind (f a) 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) : (Pmf.bind p fun a => Pmf.bind q (f a)) = Pmf.bind q fun b => Pmf.bind p fun a => f a b

@[simp]

theorem Pmf.toOuterMeasure_bind_apply {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : α → Pmf β) (s : Set β) : ↑(Pmf.toOuterMeasure (Pmf.bind p f)) s = ∑' (a : α), ↑p a * ↑(Pmf.toOuterMeasure (f a)) s

@[simp]

theorem Pmf.toMeasure_bind_apply {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : α → Pmf β) (s : Set β) [MeasurableSpace β] (hs : MeasurableSet s) : ↑↑(Pmf.toMeasure (Pmf.bind p f)) s = ∑' (a : α), ↑p a * ↑↑(Pmf.toMeasure (f a)) 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.

instance Pmf.instMonadPmf : Monad Pmf

def Pmf.bindOnSupport {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : (a : α) → a ∈ Pmf.support p → Pmf 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.bindOnSupport (fun a _ ↦ f a), see bindOnSupport_eq_bind.

@[simp]

theorem Pmf.bindOnSupport_apply {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) (b : β) : ↑(Pmf.bindOnSupport p f) b = ∑' (a : α), ↑p a * if h : ↑p a = 0 then 0 else ↑(f a h) b

@[simp]

theorem Pmf.support_bindOnSupport {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) : Pmf.support (Pmf.bindOnSupport p f) = ⋃ (a : α) (h : a ∈ Pmf.support p), Pmf.support (f a h)

theorem Pmf.mem_support_bindOnSupport_iff {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) (b : β) : b ∈ Pmf.support (Pmf.bindOnSupport p f) ↔ ∃ a h, b ∈ Pmf.support (f a h)

@[simp]

theorem Pmf.bindOnSupport_eq_bind {α : Type u_1} {β : Type u_2} (p : Pmf α) (f : α → Pmf β) : (Pmf.bindOnSupport p fun a x => f a) = Pmf.bind p f

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

theorem Pmf.bindOnSupport_eq_zero_iff {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) (b : β) : ↑(Pmf.bindOnSupport p f) b = 0 ↔ ∀ (a : α) (ha : ↑p a ≠ 0), ↑(f a ha) b = 0

@[simp]

theorem Pmf.pure_bindOnSupport {α : Type u_1} {β : Type u_2} (a : α) (f : (a' : α) → a' ∈ Pmf.support (Pmf.pure a) → Pmf β) : Pmf.bindOnSupport (Pmf.pure a) f = f a (_ : a ∈ Pmf.support (Pmf.pure a))

theorem Pmf.bindOnSupport_pure {α : Type u_1} (p : Pmf α) : (Pmf.bindOnSupport p fun a x => Pmf.pure a) = p

@[simp]

theorem Pmf.bindOnSupport_bindOnSupport {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : Pmf α) (f : (a : α) → a ∈ Pmf.support p → Pmf β) (g : (b : β) → b ∈ Pmf.support (Pmf.bindOnSupport p f) → Pmf γ) : Pmf.bindOnSupport (Pmf.bindOnSupport p f) g = Pmf.bindOnSupport p fun a ha => Pmf.bindOnSupport (f a ha) fun b hb => g b (_ : b ∈ Pmf.support (Pmf.bindOnSupport p f))

theorem Pmf.bindOnSupport_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : Pmf α) (q : Pmf β) (f : (a : α) → a ∈ Pmf.support p → (b : β) → b ∈ Pmf.support q → Pmf γ) : (Pmf.bindOnSupport p fun a ha => Pmf.bindOnSupport q (f a ha)) = Pmf.bindOnSupport q fun b hb => Pmf.bindOnSupport p fun a ha => f a ha b hb

@[simp]

theorem Pmf.toOuterMeasure_bindOnSupport_apply {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) (s : Set β) : ↑(Pmf.toOuterMeasure (Pmf.bindOnSupport p f)) s = ∑' (a : α), ↑p a * if h : ↑p a = 0 then 0 else ↑(Pmf.toOuterMeasure (f a h)) s

@[simp]

theorem Pmf.toMeasure_bindOnSupport_apply {α : Type u_1} {β : Type u_2} {p : Pmf α} (f : (a : α) → a ∈ Pmf.support p → Pmf β) (s : Set β) [MeasurableSpace β] (hs : MeasurableSet s) : ↑↑(Pmf.toMeasure (Pmf.bindOnSupport p f)) s = ∑' (a : α), ↑p a * if h : ↑p a = 0 then 0 else ↑↑(Pmf.toMeasure (f a h)) s

The measure of a set under p.bindOnSupport 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.