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.

instance Pmf.funLike {α : Type u_1} : FunLike (Pmf α) α fun x => ENNReal

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

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

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), Set.indicator s (↑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.

@[simp]

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

@[simp]

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

@[simp]

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

theorem Pmf.apply_eq_zero_iff {α : Type u_1} (p : Pmf α) (a : α) : ↑p a = 0 ↔ ¬a ∈ Pmf.support p

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

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

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

@[simp]

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

@[simp]

theorem Pmf.toOuterMeasure_apply_finset {α : Type u_1} (p : Pmf α) (s : Finset α) : ↑(Pmf.toOuterMeasure p) ↑s = Finset.sum s fun x => ↑p x

theorem Pmf.toOuterMeasure_apply_singleton {α : Type u_1} (p : Pmf α) (a : α) : ↑(Pmf.toOuterMeasure p) {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} : Pmf.toOuterMeasure p = Pmf.toOuterMeasure q ↔ p = q

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

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

@[simp]

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

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

@[simp]

theorem Pmf.toOuterMeasure_apply_fintype {α : Type u_1} (p : Pmf α) (s : Set α) [Fintype α] : ↑(Pmf.toOuterMeasure p) s = Finset.sum Finset.univ fun x => Set.indicator s (↑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.

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

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

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

theorem Pmf.toMeasure_apply_singleton {α : Type u_1} [MeasurableSpace α] (p : Pmf α) (a : α) (h : MeasurableSet {a}) : ↑↑(Pmf.toMeasure p) {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) : ↑↑(Pmf.toMeasure p) s = 0 ↔ Disjoint (Pmf.support p) s

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

@[simp]

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

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 ∩ Pmf.support p ⊆ t) : ↑↑(Pmf.toMeasure p) s ≤ ↑↑(Pmf.toMeasure p) 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 ∩ Pmf.support p = t ∩ Pmf.support p) : ↑↑(Pmf.toMeasure p) s = ↑↑(Pmf.toMeasure p) 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} : Pmf.toMeasure p = Pmf.toMeasure q ↔ p = q

@[simp]

theorem Pmf.toMeasure_apply_finset {α : Type u_1} [MeasurableSpace α] (p : Pmf α) [MeasurableSingletonClass α] (s : Finset α) : ↑↑(Pmf.toMeasure p) ↑s = Finset.sum s fun x => ↑p x

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

@[simp]

theorem Pmf.toMeasure_apply_fintype {α : Type u_1} [MeasurableSpace α] (p : Pmf α) (s : Set α) [MeasurableSingletonClass α] [Fintype α] : ↑↑(Pmf.toMeasure p) s = Finset.sum Finset.univ fun x => Set.indicator s (↑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.

theorem MeasureTheory.Measure.toPmf_apply {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (x : α) : ↑(MeasureTheory.Measure.toPmf μ) 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] : Pmf.toMeasure (MeasureTheory.Measure.toPmf μ) = μ

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

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

@[simp]

theorem Pmf.toMeasure_toPmf {α : Type u_1} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] (p : Pmf α) : MeasureTheory.Measure.toPmf (Pmf.toMeasure p) = 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] : Pmf.toMeasure p = μ ↔ p = MeasureTheory.Measure.toPmf μ

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] : MeasureTheory.Measure.toPmf μ = p ↔ μ = Pmf.toMeasure p