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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 | import OAI.Probability.MatroidSecretary.Secretary.ProbabilityModel
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Probability.Independence.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
/-!
# Reindexing a secretary rule's complete initial seed
An arbitrary finite predrawn kernel table is represented by a single finite
index. This is a measurable bijection of the complete seed, not a statistic of it.
The rule continues to read only its initial seed and the available history.
-/
open MeasureTheory
namespace MatroidProphet
/-- Reindex a secretary rule through any measurable decoding map. -/
def reindexSecretarySeed {n : ℕ} {Q Q' : Type*}
[MeasurableSpace Q] [MeasurableSpace Q']
(A : SecretaryRule n Q) (f : Q' → Q) (hf : Measurable f) : SecretaryRule n Q' where
prefixLength q := A.prefixLength (f q)
measurable_prefixLength := A.measurable_prefixLength.comp hf
decide k q h := A.decide k (f q) h
measurable_decide k := (A.measurable_decide k).comp
((hf.comp measurable_fst).prodMk measurable_snd)
/-- Canonical finite-index representation, with no nonemptiness restriction. -/
noncomputable def secretaryFinSeed {n : ℕ} {Q : Type*}
[Fintype Q] [MeasurableSpace Q] (A : SecretaryRule n Q) :
SecretaryRule n (Fin (Fintype.card Q)) :=
reindexSecretarySeed A (Fintype.equivFin Q).symm (measurable_of_finite _)
/-- Push the complete finite seed law forward along the same canonical bijection. -/
noncomputable def secretaryFinSeedLaw {Q : Type*} [Fintype Q] [MeasurableSpace Q]
(ν : Measure Q) : Measure (Fin (Fintype.card Q)) :=
ν.map (Fintype.equivFin Q)
/-- Performance against every full-seed-aware suffix adversary, on an arbitrary
ambient probability space. The seed law is fixed before the weight vector. -/
def SecretaryGuarantee.{u} {n : ℕ} {Q : Type*} [MeasurableSpace Q]
(M : Matroid (Fin n)) (A : SecretaryRule n Q) (ν : Measure Q) (c : ℝ) : Prop :=
∀ (w : Weights n), (∀ e, 0 ≤ w e) →
∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(R : Ω → Q) (σ π : Ω → ArrivalOrder n),
Measurable R → Measurable σ → Measurable π →
μ.map R = ν → μ.map σ = Secretary.uniformArrivalLaw n →
ProbabilityTheory.IndepFun R σ μ →
(∀ᵐ ω ∂μ, SamePrefix (A.prefixLength (R ω)).val (σ ω) (π ω)) →
Integrable (fun ω => secretaryReward A (R ω) w (π ω)) μ ∧
c * optimum M w ≤ ∫ ω, secretaryReward A (R ω) w (π ω) ∂μ
end MatroidProphet
end OAI
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