| import OAI.Probability.MatroidProphet.Main |
| import Mathlib.Probability.Independence.Basic |
| import Mathlib.MeasureTheory.Measure.Prod |
|
|
| namespace OAI |
|
|
| /-! Transfer a proved finite seed/prefix experiment to every ambient probability |
| space in the secretary statement. The executed adversarial order is deliberately |
| absent: no independence assumption on it is needed or introduced. -/ |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| namespace MatroidProphet |
|
|
| theorem secretary_product_law {n : ℕ} {Q Ω : Type*} |
| [MeasurableSpace Q] [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ν : Measure Q) |
| (η : Measure (ArrivalOrder n)) |
| (R : Ω → Q) (σ : Ω → ArrivalOrder n) |
| (hR : Measurable R) (hσ : Measurable σ) |
| (hRlaw : μ.map R = ν) (hσlaw : μ.map σ = η) |
| (hind : IndepFun R σ μ) : |
| MeasurePreserving (fun ω => (R ω, σ ω)) μ (ν.prod (η)) := by |
| refine ⟨hR.prodMk hσ, ?_⟩ |
| rw [hind.map_prod_eq_prod_map_map hR.aemeasurable hσ.aemeasurable, hRlaw, hσlaw] |
|
|
| / |
| ambient space once that law is established on the independent product model. -/ |
| theorem secretary_reconstruction_law {n bits : ℕ} {Q Ω : Type*} |
| [MeasurableSpace Q] [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ν : Measure Q) |
| (η : Measure (ArrivalOrder n)) |
| (ρ : Measure (Seed bits)) (reconstruct : Q × ArrivalOrder n → Seed bits) |
| (hm : Measurable reconstruct) |
| (hlaw : (ν.prod (η)).map reconstruct = ρ) |
| (R : Ω → Q) (σ : Ω → ArrivalOrder n) |
| (hR : Measurable R) (hσ : Measurable σ) |
| (hRlaw : μ.map R = ν) (hσlaw : μ.map σ = η) |
| (hind : IndepFun R σ μ) : |
| MeasurePreserving (fun ω => reconstruct (R ω, σ ω)) μ ρ := by |
| have hbase := secretary_product_law μ ν η R σ hR hσ hRlaw hσlaw hind |
| exact (show MeasurePreserving reconstruct (ν.prod (η)) ρ |
| from ⟨hm, hlaw⟩).comp hbase |
|
|
| / |
| support, pull back jointly; knowing only a mask marginal would not suffice. -/ |
| theorem secretary_product_ae {n : ℕ} {Q Ω : Type*} |
| [MeasurableSpace Q] [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (ν : Measure Q) |
| (η : Measure (ArrivalOrder n)) |
| (R : Ω → Q) (σ : Ω → ArrivalOrder n) |
| (hR : Measurable R) (hσ : Measurable σ) |
| (hRlaw : μ.map R = ν) (hσlaw : μ.map σ = η) |
| (hind : IndepFun R σ μ) (P : Q × ArrivalOrder n → Prop) |
| (hP : ∀ᵐ x ∂ν.prod (η), P x) : |
| ∀ᵐ ω ∂μ, P (R ω, σ ω) := |
| (secretary_product_law μ ν η R σ hR hσ hRlaw hσlaw hind).quasiMeasurePreserving.ae hP |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
