# mathlibdocumentation

measure_theory.probability_mass_function.constructions

# Specific Constructions of Probability Mass Functions #

This file gives a number of different pmf constructions for common probability distributions.

map and seq allow pushing a pmf α along a function f : α → β (or distribution of functions f : pmf (α → β)) to get a pmf β

of_finset and of_fintype simplify the construction of a pmf α from a function f : α → ℝ≥0, by allowing the "sum equals 1" constraint to be in terms of finset.sum instead of tsum. of_multiset, uniform_of_finset, and uniform_of_fintype construct probability mass functions from the corresponding object, with proportional weighting for each element of the object.

normalize constructs a pmf α by normalizing a function f : α → ℝ≥0 by its sum, and filter uses this to filter the support of a pmf and re-normalize the new distribution.

bernoulli represents the bernoulli distribution on bool

noncomputable def pmf.map {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) :
pmf β

The functorial action of a function on a pmf.

Equations
@[simp]
theorem pmf.map_apply {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) (b : β) :
(pmf.map f p) b = ∑' (a : α), ite (b = f a) (p a) 0
@[simp]
theorem pmf.support_map {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) :
theorem pmf.mem_support_map_iff {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) (b : β) :
b (pmf.map f p).support ∃ (a : α) (H : a p.support), f a = b
theorem pmf.bind_pure_comp {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) :
p.bind (pmf.pure f) = p
theorem pmf.map_id {α : Type u_1} (p : pmf α) :
= p
theorem pmf.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (p : pmf α) (g : β → γ) :
(pmf.map f p) = pmf.map (g f) p
theorem pmf.pure_map {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) :
(pmf.pure a) = pmf.pure (f a)
@[simp]
theorem pmf.to_outer_measure_map_apply {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) (s : set β) :
@[simp]
theorem pmf.to_measure_map_apply {α : Type u_1} {β : Type u_2} (f : α → β) (p : pmf α) (s : set β) (hf : measurable f) (hs : measurable_set s) :
noncomputable def pmf.seq {α : Type u_1} {β : Type u_2} (q : pmf (α → β)) (p : pmf α) :
pmf β

The monadic sequencing operation for pmf.

Equations
@[simp]
theorem pmf.seq_apply {α : Type u_1} {β : Type u_2} (q : pmf (α → β)) (p : pmf α) (b : β) :
(q.seq p) b = ∑' (f : α → β) (a : α), ite (b = f a) (q f * p a) 0
@[simp]
theorem pmf.support_seq {α : Type u_1} {β : Type u_2} (q : pmf (α → β)) (p : pmf α) :
(q.seq p).support = ⋃ (f : α → β) (H : f q.support), f '' p.support
theorem pmf.mem_support_seq_iff {α : Type u_1} {β : Type u_2} (q : pmf (α → β)) (p : pmf α) (b : β) :
b (q.seq p).support ∃ (f : α → β) (H : f q.support), b f '' p.support
def pmf.of_finset {α : Type u_1} (f : α → nnreal) (s : finset α) (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) :
pmf α

Given a finset s and a function f : α → ℝ≥0 with sum 1 on s, such that f a = 0 for a ∉ s, we get a pmf

Equations
@[simp]
theorem pmf.of_finset_apply {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) (a : α) :
s h h') a = f a
@[simp]
theorem pmf.support_of_finset {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) :
s h h').support =
theorem pmf.mem_support_of_finset_iff {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) (a : α) :
a s h h').support a s f a 0
theorem pmf.of_finset_apply_of_not_mem {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) {a : α} (ha : a s) :
s h h') a = 0
@[simp]
theorem pmf.to_outer_measure_of_finset_apply {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) (t : set α) :
s h h').to_outer_measure) t = ∑' (x : α), t.indicator f x
@[simp]
theorem pmf.to_measure_of_finset_apply {α : Type u_1} {f : α → nnreal} {s : finset α} (h : s.sum (λ (a : α), f a) = 1) (h' : ∀ (a : α), a sf a = 0) (t : set α) (ht : measurable_set t) :
s h h').to_measure) t = ∑' (x : α), t.indicator f x
def pmf.of_fintype {α : Type u_1} [fintype α] (f : α → nnreal) (h : finset.univ.sum (λ (a : α), f a) = 1) :
pmf α

Given a finite type α and a function f : α → ℝ≥0 with sum 1, we get a pmf.

Equations
• = _
@[simp]
theorem pmf.of_fintype_apply {α : Type u_1} [fintype α] {f : α → nnreal} (h : finset.univ.sum (λ (a : α), f a) = 1) (a : α) :
h) a = f a
@[simp]
theorem pmf.support_of_fintype {α : Type u_1} [fintype α] {f : α → nnreal} (h : finset.univ.sum (λ (a : α), f a) = 1) :
theorem pmf.mem_support_of_fintype_iff {α : Type u_1} [fintype α] {f : α → nnreal} (h : finset.univ.sum (λ (a : α), f a) = 1) (a : α) :
a h).support f a 0
@[simp]
theorem pmf.to_outer_measure_of_fintype_apply {α : Type u_1} [fintype α] {f : α → nnreal} (h : finset.univ.sum (λ (a : α), f a) = 1) (s : set α) :
h).to_outer_measure) s = ∑' (x : α), s.indicator f x
@[simp]
theorem pmf.to_measure_of_fintype_apply {α : Type u_1} [fintype α] {f : α → nnreal} (h : finset.univ.sum (λ (a : α), f a) = 1) (s : set α) (hs : measurable_set s) :
h).to_measure) s = ∑' (x : α), s.indicator f x
noncomputable def pmf.of_multiset {α : Type u_1} (s : multiset α) (hs : s 0) :
pmf α

Given a non-empty multiset s we construct the pmf which sends a to the fraction of elements in s that are a.

Equations
@[simp]
theorem pmf.of_multiset_apply {α : Type u_1} {s : multiset α} (hs : s 0) (a : α) :
hs) a = s) /
@[simp]
theorem pmf.support_of_multiset {α : Type u_1} {s : multiset α} (hs : s 0) :
theorem pmf.mem_support_of_multiset_iff {α : Type u_1} {s : multiset α} (hs : s 0) (a : α) :
theorem pmf.of_multiset_apply_of_not_mem {α : Type u_1} {s : multiset α} (hs : s 0) {a : α} (ha : a s) :
hs) a = 0
@[simp]
theorem pmf.to_outer_measure_of_multiset_apply {α : Type u_1} {s : multiset α} (hs : s 0) (t : set α) :
hs).to_outer_measure) t = (∑' (x : α), (multiset.filter (λ (_x : α), _x t) s))) /
@[simp]
theorem pmf.to_measure_of_multiset_apply {α : Type u_1} {s : multiset α} (hs : s 0) (t : set α) (ht : measurable_set t) :
hs).to_measure) t = (∑' (x : α), (multiset.filter (λ (_x : α), _x t) s))) /
noncomputable def pmf.uniform_of_finset {α : Type u_1} (s : finset α) (hs : s.nonempty) :
pmf α

Uniform distribution taking the same non-zero probability on the nonempty finset s

Equations
@[simp]
theorem pmf.uniform_of_finset_apply {α : Type u_1} {s : finset α} (hs : s.nonempty) (a : α) :
hs) a = ite (a s) ((s.card))⁻¹ 0
theorem pmf.uniform_of_finset_apply_of_mem {α : Type u_1} {s : finset α} (hs : s.nonempty) {a : α} (ha : a s) :
hs) a = ((s.card))⁻¹
theorem pmf.uniform_of_finset_apply_of_not_mem {α : Type u_1} {s : finset α} (hs : s.nonempty) {a : α} (ha : a s) :
hs) a = 0
@[simp]
theorem pmf.support_uniform_of_finset {α : Type u_1} {s : finset α} (hs : s.nonempty) :
hs).support = s
theorem pmf.mem_support_uniform_of_finset_iff {α : Type u_1} {s : finset α} (hs : s.nonempty) (a : α) :
a hs).support a s
@[simp]
theorem pmf.to_outer_measure_uniform_of_finset_apply {α : Type u_1} {s : finset α} (hs : s.nonempty) (t : set α) :
hs).to_outer_measure) t = ((finset.filter (λ (_x : α), _x t) s).card) / (s.card)
@[simp]
theorem pmf.to_measure_uniform_of_finset_apply {α : Type u_1} {s : finset α} (hs : s.nonempty) (t : set α) (ht : measurable_set t) :
hs).to_measure) t = ((finset.filter (λ (_x : α), _x t) s).card) / (s.card)
noncomputable def pmf.uniform_of_fintype (α : Type u_1) [fintype α] [nonempty α] :
pmf α

The uniform pmf taking the same uniform value on all of the fintype α

Equations
@[simp]
theorem pmf.uniform_of_fintype_apply {α : Type u_1} [fintype α] [nonempty α] (a : α) :
@[simp]
theorem pmf.support_uniform_of_fintype (α : Type u_1) [fintype α] [nonempty α] :
theorem pmf.mem_support_uniform_of_fintype {α : Type u_1} [fintype α] [nonempty α] (a : α) :
theorem pmf.to_outer_measure_uniform_of_fintype_apply {α : Type u_1} [fintype α] [nonempty α] (s : set α) :
theorem pmf.to_measure_uniform_of_fintype_apply {α : Type u_1} [fintype α] [nonempty α] (s : set α) (hs : measurable_set s) :
noncomputable def pmf.normalize {α : Type u_1} (f : α → nnreal) (hf0 : tsum f 0) :
pmf α

Given a f with non-zero sum, we get a pmf by normalizing f by it's tsum

Equations
• hf0 = λ (a : α), f a * (∑' (x : α), f x)⁻¹, _⟩
@[simp]
theorem pmf.normalize_apply {α : Type u_1} {f : α → nnreal} (hf0 : tsum f 0) (a : α) :
hf0) a = f a * (∑' (x : α), f x)⁻¹
@[simp]
theorem pmf.support_normalize {α : Type u_1} {f : α → nnreal} (hf0 : tsum f 0) :
hf0).support =
theorem pmf.mem_support_normalize_iff {α : Type u_1} {f : α → nnreal} (hf0 : tsum f 0) (a : α) :
a hf0).support f a 0
noncomputable def pmf.filter {α : Type u_1} (p : pmf α) (s : set α) (h : ∃ (a : α) (H : a s), a p.support) :
pmf α

Create new pmf by filtering on a set with non-zero measure and normalizing

Equations
@[simp]
theorem pmf.filter_apply {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) (a : α) :
(p.filter s h) a = s.indicator p a * (∑' (a' : α), s.indicator p a')⁻¹
theorem pmf.filter_apply_eq_zero_of_not_mem {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) {a : α} (ha : a s) :
(p.filter s h) a = 0
theorem pmf.mem_support_filter_iff {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) {a : α} :
a (p.filter s h).support a s a p.support
@[simp]
theorem pmf.support_filter {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) :
(p.filter s h).support = s p.support
theorem pmf.filter_apply_eq_zero_iff {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) (a : α) :
(p.filter s h) a = 0 a s a p.support
theorem pmf.filter_apply_ne_zero_iff {α : Type u_1} {p : pmf α} {s : set α} (h : ∃ (a : α) (H : a s), a p.support) (a : α) :
(p.filter s h) a 0 a s a p.support
noncomputable def pmf.bernoulli (p : nnreal) (h : p 1) :

A pmf which assigns probability p to tt and 1 - p to ff.

Equations
@[simp]
theorem pmf.bernoulli_apply {p : nnreal} (h : p 1) (b : bool) :
h) b = cond b p (1 - p)
@[simp]
theorem pmf.support_bernoulli {p : nnreal} (h : p 1) :
h).support = {b : bool | cond b (p 0) (p 1)}
theorem pmf.mem_support_bernoulli_iff {p : nnreal} (h : p 1) (b : bool) :
b h).support cond b (p 0) (p 1)