| 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 |
|
|
