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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Probability.MatroidProphet.Main
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Tactic.FieldSimp
import Mathlib.Probability.Distributions.Binomial
import Mathlib.Probability.ProbabilityMassFunction.Integrals
import Mathlib.Probability.Distributions.Uniform
namespace OAI
namespace MatroidProphet.SecretaryBinomialMixture
open Finset
open scoped NNReal
variable {α : Type*} [DecidableEq α]
/-- A homogeneous Bernoulli mask has the exact cardinality-dependent mass. -/
lemma bitsWeight_const (p : ℝ) (V S : Finset α) (hS : S ⊆ V) :
bitsWeight (fun _ => p) V S = p ^ S.card * (1 - p) ^ (V.card - S.card) := by
classical
unfold bitsWeight
rw [Finset.prod_ite]
have hyes : V.filter (fun e => e ∈ S) = S := by
ext e
simp only [mem_filter]
exact ⟨fun h => h.2, fun h => ⟨hS h, h⟩⟩
have hno : V.filter (fun e => e ∉ S) = V \ S := by
ext e
simp
rw [hyes, hno, prod_const, prod_const, card_sdiff_of_subset hS]
/-- The cardinality of a predrawn product mask has the full binomial law.
The identity also holds at `p = 0`, `p = 1`, and for an empty ground set. -/
theorem bitsExpectation_card (p : ℝ) (V : Finset α) (f : ℕ → ℝ) :
bitsExpectation (fun _ => p) V (fun S => f S.card) =
∑ k ∈ range (V.card + 1),
(V.card.choose k : ℝ) * p ^ k * (1 - p) ^ (V.card - k) * f k := by
classical
unfold bitsExpectation
rw [sum_powerset]
apply sum_congr rfl
intro k hk
calc
(∑ S ∈ V.powersetCard k, bitsWeight (fun _ => p) V S * f S.card) =
∑ _S ∈ V.powersetCard k, p ^ k * (1 - p) ^ (V.card - k) * f k := by
apply sum_congr rfl
intro S hS
obtain ⟨hSV, hcard⟩ := mem_powersetCard.mp hS
rw [bitsWeight_const p V S hSV, hcard]
_ = (V.card.choose k : ℝ) * p ^ k * (1 - p) ^ (V.card - k) * f k := by
rw [sum_const, card_powersetCard, nsmul_eq_mul]
ring
/-- Complete averaging over the binomial prefix length cancels the uniform
fixed-size subset normalizer and recovers the product mask expectation. -/
theorem binomial_uniform_subset_mixture (p : ℝ) (V : Finset α)
(f : Finset α → ℝ) :
(∑ k ∈ range (V.card + 1),
((V.card.choose k : ℝ) * p ^ k * (1 - p) ^ (V.card - k)) *
((V.card.choose k : ℝ)⁻¹ * ∑ S ∈ V.powersetCard k, f S)) =
bitsExpectation (fun _ => p) V f := by
classical
unfold bitsExpectation
rw [sum_powerset]
apply sum_congr rfl
intro k hk
have hkV : k ≤ V.card := Nat.lt_succ_iff.mp (mem_range.mp hk)
have hc : (V.card.choose k : ℝ) ≠ 0 := by
exact_mod_cast Nat.ne_of_gt (Nat.choose_pos hkV)
calc
((V.card.choose k : ℝ) * p ^ k * (1 - p) ^ (V.card - k)) *
((V.card.choose k : ℝ)⁻¹ * ∑ S ∈ V.powersetCard k, f S) =
(p ^ k * (1 - p) ^ (V.card - k)) * ∑ S ∈ V.powersetCard k, f S := by
field_simp
_ = ∑ S ∈ V.powersetCard k, bitsWeight (fun _ => p) V S * f S := by
rw [mul_sum]
apply sum_congr rfl
intro S hS
obtain ⟨hSV, hcard⟩ := mem_powersetCard.mp hS
rw [bitsWeight_const p V S hSV, hcard]
/-- Equivalent implementation: predraw a product mask, keep only its cardinality,
then draw a uniform subset of that cardinality independently. -/
theorem predrawn_card_uniform_subset (p : ℝ) (V : Finset α)
(f : Finset α → ℝ) :
bitsExpectation (fun _ => p) V
(fun B => (V.card.choose B.card : ℝ)⁻¹ *
∑ S ∈ V.powersetCard B.card, f S) =
bitsExpectation (fun _ => p) V f := by
rw [bitsExpectation_card p V (fun k => (V.card.choose k : ℝ)⁻¹ *
∑ S ∈ V.powersetCard k, f S)]
exact binomial_uniform_subset_mixture p V f
open MeasureTheory
/-- The finite binomial law used for the initial prefix-length draw, with its
actual integral rather than an unnormalized formal sum. -/
theorem integral_binomial (p : ℝ≥0) (hp : p ≤ 1) (n : ℕ) (f : ℕ → ℝ) :
(∫ k, f k ∂ProbabilityTheory.binomial n
⟨p, p.coe_nonneg, by exact_mod_cast hp⟩) =
∑ k ∈ range (n + 1),
(n.choose k : ℝ) * (p : ℝ) ^ k * (1 - (p : ℝ)) ^ (n - k) * f k := by
simpa only [smul_eq_mul, ← Nat.range_succ_eq_Iic] using
(ProbabilityTheory.integral_binomial (n := n)
(p := ⟨p, p.coe_nonneg, by exact_mod_cast hp⟩) f)
/-- Sampling the finite binomial length and then a uniform fixed-cardinality
subset gives the homogeneous Bernoulli product mask exactly. -/
theorem integral_binomial_uniform_subset (p : ℝ≥0) (hp : p ≤ 1)
(V : Finset α) (f : Finset α → ℝ) :
(∫ k, (V.card.choose k : ℝ)⁻¹ * ∑ S ∈ V.powersetCard k, f S
∂ProbabilityTheory.binomial V.card ⟨p, p.coe_nonneg, by exact_mod_cast hp⟩) =
bitsExpectation (fun _ => (p : ℝ)) V f := by
rw [integral_binomial p hp V.card (fun k => (V.card.choose k : ℝ)⁻¹ *
∑ S ∈ V.powersetCard k, f S)]
exact binomial_uniform_subset_mixture (p : ℝ) V f
/-- Conversion from the actual finite uniform law to a normalized sum. -/
theorem integral_uniform_fintype {β : Type*} [Fintype β] [Nonempty β]
[MeasurableSpace β] [MeasurableSingletonClass β] (f : β → ℝ) :
(∫ b, f b ∂(PMF.uniformOfFintype β).toMeasure) =
(Fintype.card β : ℝ)⁻¹ * ∑ b, f b := by
rw [PMF.integral_eq_sum, mul_sum]
apply sum_congr rfl
intro b hb
simp only [PMF.uniformOfFintype_apply, ENNReal.toReal_inv,
ENNReal.toReal_natCast, smul_eq_mul]
end MatroidProphet.SecretaryBinomialMixture
end OAI
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