Probability mass functions

This file is about probability mass functions or discrete probability measures: a function α → ℝ≥0∞ such that the values have (infinite) sum 1.

Construction of monadic pure and bind is found in ProbabilityMassFunction/Monad.lean, other constructions of PMFs are found in ProbabilityMassFunction/Constructions.lean.

Given p : PMF α, PMF.toOuterMeasure constructs an OuterMeasure on α, by assigning each set the sum of the probabilities of each of its elements. Under this outer measure, every set is Carathéodory-measurable, so we can further extend this to a Measure on α, see PMF.toMeasure. PMF.toMeasure.isProbabilityMeasure shows this associated measure is a probability measure. Conversely, given a probability measure μ on a measurable space α with all singleton sets measurable, μ.toPMF constructs a PMF on α, setting the probability mass of a point x to be the measure of the singleton set {x}.

Tags

probability mass function, discrete probability measure

def PMF (α : Type u) : Type u

A probability mass function, or discrete probability measures is a function α → ℝ≥0∞ such that the values have (infinite) sum 1.

Equations

• PMF α = { f : α → ENNReal // HasSum f 1 }

instance PMF.instFunLike {α : Type u_1} : FunLike (PMF α) α ENNReal

Equations

• PMF.instFunLike = { coe := fun (p : PMF α) (a : α) => ↑p a, coe_injective' := ⋯ }

theorem PMF.ext_iff {α : Type u_1} {p : PMF α} {q : PMF α} : p = q ↔ ∀ (x : α), p x = q x

theorem PMF.ext {α : Type u_1} {p : PMF α} {q : PMF α} (h : ∀ (x : α), p x = q x) : p = q

theorem PMF.hasSum_coe_one {α : Type u_1} (p : PMF α) : HasSum (⇑p) 1

@[simp]

theorem PMF.tsum_coe {α : Type u_1} (p : PMF α) : ∑' (a : α), p a = 1

theorem PMF.tsum_coe_ne_top {α : Type u_1} (p : PMF α) : ∑' (a : α), p a ≠ ⊤

theorem PMF.tsum_coe_indicator_ne_top {α : Type u_1} (p : PMF α) (s : Set α) : ∑' (a : α), s.indicator (⇑p) a ≠ ⊤

@[simp]

theorem PMF.coe_ne_zero {α : Type u_1} (p : PMF α) : ⇑p ≠ 0

def PMF.support {α : Type u_1} (p : PMF α) : Set α

The support of a PMF is the set where it is nonzero.

Equations

• p.support = Function.support ⇑p

@[simp]

theorem PMF.mem_support_iff {α : Type u_1} (p : PMF α) (a : α) : a ∈ p.support ↔ p a ≠ 0

@[simp]

theorem PMF.support_nonempty {α : Type u_1} (p : PMF α) : p.support.Nonempty

@[simp]

theorem PMF.support_countable {α : Type u_1} (p : PMF α) : p.support.Countable

theorem PMF.apply_eq_zero_iff {α : Type u_1} (p : PMF α) (a : α) : p a = 0 ↔ a ∉ p.support

theorem PMF.apply_pos_iff {α : Type u_1} (p : PMF α) (a : α) : 0 < p a ↔ a ∈ p.support

theorem PMF.apply_eq_one_iff {α : Type u_1} (p : PMF α) (a : α) : p a = 1 ↔ p.support = {a}

theorem PMF.coe_le_one {α : Type u_1} (p : PMF α) (a : α) : p a ≤ 1

theorem PMF.apply_ne_top {α : Type u_1} (p : PMF α) (a : α) : p a ≠ ⊤

theorem PMF.apply_lt_top {α : Type u_1} (p : PMF α) (a : α) : p a < ⊤

def PMF.toOuterMeasure {α : Type u_1} (p : PMF α) : MeasureTheory.OuterMeasure α

Construct an OuterMeasure from a PMF, by assigning measure to each set s : Set α equal to the sum of p x for each x ∈ α.

Equations

• p.toOuterMeasure = MeasureTheory.OuterMeasure.sum fun (x : α) => p x • MeasureTheory.OuterMeasure.dirac x

theorem PMF.toOuterMeasure_apply {α : Type u_1} (p : PMF α) (s : Set α) : p.toOuterMeasure s = ∑' (x : α), s.indicator (⇑p) x

@[simp]

theorem PMF.toOuterMeasure_caratheodory {α : Type u_1} (p : PMF α) : p.toOuterMeasure.caratheodory = ⊤

@[simp]

theorem PMF.toOuterMeasure_apply_finset {α : Type u_1} (p : PMF α) (s : Finset α) : p.toOuterMeasure ↑s = ∑ x ∈ s, p x

theorem PMF.toOuterMeasure_apply_singleton {α : Type u_1} (p : PMF α) (a : α) : p.toOuterMeasure {a} = p a

theorem PMF.toOuterMeasure_injective {α : Type u_1} : Function.Injective PMF.toOuterMeasure

@[simp]

theorem PMF.toOuterMeasure_inj {α : Type u_1} {p : PMF α} {q : PMF α} : p.toOuterMeasure = q.toOuterMeasure ↔ p = q

theorem PMF.toOuterMeasure_apply_eq_zero_iff {α : Type u_1} (p : PMF α) (s : Set α) : p.toOuterMeasure s = 0 ↔ Disjoint p.support s

theorem PMF.toOuterMeasure_apply_eq_one_iff {α : Type u_1} (p : PMF α) (s : Set α) : p.toOuterMeasure s = 1 ↔ p.support ⊆ s

@[simp]

theorem PMF.toOuterMeasure_apply_inter_support {α : Type u_1} (p : PMF α) (s : Set α) : p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s

theorem PMF.toOuterMeasure_mono {α : Type u_1} (p : PMF α) {s : Set α} {t : Set α} (h : s ∩ p.support ⊆ t) : p.toOuterMeasure s ≤ p.toOuterMeasure t

Slightly stronger than OuterMeasure.mono having an intersection with p.support.

theorem PMF.toOuterMeasure_apply_eq_of_inter_support_eq {α : Type u_1} (p : PMF α) {s : Set α} {t : Set α} (h : s ∩ p.support = t ∩ p.support) : p.toOuterMeasure s = p.toOuterMeasure t

@[simp]

theorem PMF.toOuterMeasure_apply_fintype {α : Type u_1} (p : PMF α) (s : Set α) [Fintype α] : p.toOuterMeasure s = ∑ x : α, s.indicator (⇑p) x

def PMF.toMeasure {α : Type u_1} [MeasurableSpace α] (p : PMF α) : MeasureTheory.Measure α

Since every set is Carathéodory-measurable under PMF.toOuterMeasure, we can further extend this OuterMeasure to a Measure on α.

Equations

• p.toMeasure = p.toOuterMeasure.toMeasure ⋯

theorem PMF.toOuterMeasure_apply_le_toMeasure_apply {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) : p.toOuterMeasure s ≤ p.toMeasure s

theorem PMF.toMeasure_apply_eq_toOuterMeasure_apply {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) : p.toMeasure s = p.toOuterMeasure s

theorem PMF.toMeasure_apply {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) : p.toMeasure s = ∑' (x : α), s.indicator (⇑p) x

theorem PMF.toMeasure_apply_singleton {α : Type u_1} [MeasurableSpace α] (p : PMF α) (a : α) (h : MeasurableSet {a}) : p.toMeasure {a} = p a

theorem PMF.toMeasure_apply_eq_zero_iff {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) : p.toMeasure s = 0 ↔ Disjoint p.support s

theorem PMF.toMeasure_apply_eq_one_iff {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) : p.toMeasure s = 1 ↔ p.support ⊆ s

@[simp]

theorem PMF.toMeasure_apply_inter_support {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) (hs : MeasurableSet s) (hp : MeasurableSet p.support) : p.toMeasure (s ∩ p.support) = p.toMeasure s

@[simp]

theorem PMF.restrict_toMeasure_support {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) : p.toMeasure.restrict p.support = p.toMeasure

theorem PMF.toMeasure_mono {α : Type u_1} [MeasurableSpace α] (p : PMF α) {s : Set α} {t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (h : s ∩ p.support ⊆ t) : p.toMeasure s ≤ p.toMeasure t

theorem PMF.toMeasure_apply_eq_of_inter_support_eq {α : Type u_1} [MeasurableSpace α] (p : PMF α) {s : Set α} {t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (h : s ∩ p.support = t ∩ p.support) : p.toMeasure s = p.toMeasure t

theorem PMF.toMeasure_injective {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] : Function.Injective PMF.toMeasure

@[simp]

theorem PMF.toMeasure_inj {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {p : PMF α} {q : PMF α} : p.toMeasure = q.toMeasure ↔ p = q

@[simp]

theorem PMF.toMeasure_apply_finset {α : Type u_1} [MeasurableSpace α] (p : PMF α) [MeasurableSingletonClass α] (s : Finset α) : p.toMeasure ↑s = ∑ x ∈ s, p x

theorem PMF.toMeasure_apply_of_finite {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) [MeasurableSingletonClass α] (hs : s.Finite) : p.toMeasure s = ∑' (x : α), s.indicator (⇑p) x

@[simp]

theorem PMF.toMeasure_apply_fintype {α : Type u_1} [MeasurableSpace α] (p : PMF α) (s : Set α) [MeasurableSingletonClass α] [Fintype α] : p.toMeasure s = ∑ x : α, s.indicator (⇑p) x

def MeasureTheory.Measure.toPMF {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [h : MeasureTheory.IsProbabilityMeasure μ] : PMF α

Given that α is a countable, measurable space with all singleton sets measurable, we can convert any probability measure into a PMF, where the mass of a point is the measure of the singleton set under the original measure.

Equations

• μ.toPMF = ⟨fun (x : α) => μ {x}, ⋯⟩

theorem MeasureTheory.Measure.toPMF_apply {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (x : α) : μ.toPMF x = μ {x}

@[simp]

theorem MeasureTheory.Measure.toPMF_toMeasure {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] : μ.toPMF.toMeasure = μ

instance PMF.toMeasure.isProbabilityMeasure {α : Type u_1} [MeasurableSpace α] (p : PMF α) : MeasureTheory.IsProbabilityMeasure p.toMeasure

The measure associated to a PMF by toMeasure is a probability measure.

Equations

• ⋯ = ⋯

@[simp]

theorem PMF.toMeasure_toPMF {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) : p.toMeasure.toPMF = p

theorem PMF.toMeasure_eq_iff_eq_toPMF {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] : p.toMeasure = μ ↔ p = μ.toPMF

theorem PMF.toPMF_eq_iff_toMeasure_eq {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : PMF α) (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] : μ.toPMF = p ↔ μ = p.toMeasure