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.

Equations

• PMF.pure a = ⟨fun (a' : α) => if a' = a then 1 else 0, ⋯⟩

@[simp]

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

@[simp]

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

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

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

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

instance PMF.instInhabited {α : Type u_1} [Inhabited α] : Inhabited (PMF α)

Equations

• PMF.instInhabited = { default := PMF.pure default }

@[simp]

theorem PMF.toOuterMeasure_pure_apply {α : Type u_1} (a : α) (s : Set α) : (pure a).toOuterMeasure 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) : (pure a).toMeasure 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] : (pure a).toMeasure = MeasureTheory.Measure.dirac a

@[simp]

theorem PMF.toPMF_dirac {α : Type u_1} (a : α) [MeasurableSpace α] [Countable α] [h : MeasurableSingletonClass α] : (MeasureTheory.Measure.dirac a).toPMF = 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.

Equations

• p.bind f = ⟨fun (b : β) => ∑' (a : α), p a * (f a) b, ⋯⟩

@[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 = ⋃ a ∈ p.support, (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 ∈ p.support, b ∈ (f a).support

@[simp]

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

@[simp]

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

@[simp]

theorem PMF.bind_const {α : Type u_1} {β : Type u_2} (p : PMF α) (q : PMF β) : (p.bind fun (x : α) => 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) : (p.bind f).bind g = p.bind fun (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 fun (a : α) => q.bind (f a)) = q.bind fun (b : β) => p.bind fun (a : α) => 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) : (p.bind f).toOuterMeasure s = ∑' (a : α), p a * (f a).toOuterMeasure 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) : (p.bind f).toMeasure s = ∑' (a : α), p a * (f a).toMeasure 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.instMonad : Monad PMF

Equations

• One or more equations did not get rendered due to their size.

def PMF.bindOnSupport {α : Type u_1} {β : Type u_2} (p : PMF α) (f : (a : α) → a ∈ p.support → 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.

Equations

• p.bindOnSupport f = ⟨fun (b : β) => ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h) b, ⋯⟩

@[simp]

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

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

@[simp]

theorem PMF.bindOnSupport_eq_bind {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β) : (p.bindOnSupport fun (a : α) (x : a ∈ p.support) => f a) = p.bind 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 ∈ p.support → PMF β) (b : β) : (p.bindOnSupport 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' ∈ (pure a).support → PMF β) : (pure a).bindOnSupport f = f a ⋯

theorem PMF.bindOnSupport_pure {α : Type u_1} (p : PMF α) : (p.bindOnSupport fun (a : α) (x : a ∈ p.support) => 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 ∈ p.support → PMF β) (g : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ) : (p.bindOnSupport f).bindOnSupport g = p.bindOnSupport fun (a : α) (ha : a ∈ p.support) => (f a ha).bindOnSupport fun (b : β) (hb : b ∈ (f a ha).support) => g b ⋯

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 ∈ p.support → (b : β) → b ∈ q.support → PMF γ) : (p.bindOnSupport fun (a : α) (ha : a ∈ p.support) => q.bindOnSupport (f a ha)) = q.bindOnSupport fun (b : β) (hb : b ∈ q.support) => p.bindOnSupport fun (a : α) (ha : a ∈ p.support) => 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 ∈ p.support → PMF β) (s : Set β) : (p.bindOnSupport f).toOuterMeasure s = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s

@[simp]

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