| import OAI.Probability.MatroidSecretary.Sampling.FiniteSeedModel |
| import Mathlib.MeasureTheory.Measure.Map |
| import Mathlib.MeasureTheory.Integral.Bochner.Basic |
| import Mathlib.Probability.Independence.Basic |
|
|
| namespace OAI |
|
|
| /-! Exact finite-seed encoding, measurability, and law preservation. -/ |
|
|
| open MeasureTheory |
| namespace MatroidProphet |
|
|
| variable (α : Type*) [Fintype α] |
|
|
| lemma finiteSeedEncode_injective : Function.Injective (finiteSeedEncode α) := by |
| classical |
| intro a b h |
| have hb : finiteSeedEncode α b ((Fintype.equivFin α) a) = true := by |
| rw [← h] |
| simp [finiteSeedEncode] |
| have he : (Fintype.equivFin α) b = (Fintype.equivFin α) a := by |
| simpa [finiteSeedEncode] using hb |
| exact (Fintype.equivFin α).injective he.symm |
|
|
| variable [Nonempty α] |
|
|
| @[simp] lemma finiteSeedDecode_encode (a : α) : |
| finiteSeedDecode α (finiteSeedEncode α a) = a := by |
| exact Function.leftInverse_invFun (finiteSeedEncode_injective α) a |
|
|
| variable [MeasurableSpace α] [MeasurableSingletonClass α] |
|
|
| omit [Nonempty α] in |
| lemma measurable_finiteSeedEncode : Measurable (finiteSeedEncode α) := |
| measurable_of_finite _ |
|
|
| omit [MeasurableSingletonClass α] in |
| lemma measurable_finiteSeedDecode : Measurable (finiteSeedDecode α) := |
| measurable_of_finite _ |
|
|
| /-- The full original law is recovered, not merely the law of a used statistic. -/ |
| theorem finiteSeed_map_roundtrip (μ : Measure α) : |
| (μ.map (finiteSeedEncode α)).map (finiteSeedDecode α) = μ := by |
| rw [Measure.map_map (measurable_finiteSeedDecode α) (measurable_finiteSeedEncode α)] |
| simp only [Function.comp_def, finiteSeedDecode_encode] |
| exact Measure.map_id |
|
|
| /-- Decoding the encoded law preserves every event of the original seed. -/ |
| theorem finiteSeed_decode_measurePreserving (μ : Measure α) : |
| MeasurePreserving (finiteSeedDecode α) (μ.map (finiteSeedEncode α)) μ := |
| ⟨measurable_finiteSeedDecode α, finiteSeed_map_roundtrip α μ⟩ |
|
|
| /-- Arbitrary real payoff statistics have exactly the same expectation. -/ |
| theorem finiteSeed_integral_decode (μ : Measure α) (g : α → ℝ) : |
| (∫ r, g (finiteSeedDecode α r) ∂μ.map (finiteSeedEncode α)) = ∫ a, g a ∂μ := by |
| have h := integral_map_of_stronglyMeasurable (μ := μ) |
| (measurable_finiteSeedEncode α) |
| ((measurable_of_finite (fun r => g (finiteSeedDecode α r))).stronglyMeasurable) |
| simpa only [finiteSeedDecode_encode] using h |
|
|
| /-- Encoding a full seed does not change its independence from any other data. -/ |
| theorem finiteSeed_indepFun_iff {Ω β : Type*} [MeasurableSpace Ω] [MeasurableSpace β] |
| (μ : Measure Ω) (X : Ω → β) (R : Ω → α) : |
| ProbabilityTheory.IndepFun X (finiteSeedEncode α ∘ R) μ ↔ |
| ProbabilityTheory.IndepFun X R μ := by |
| constructor |
| · intro h |
| have hd := h.comp measurable_id (measurable_finiteSeedDecode α) |
| simpa only [Function.comp_def, id_eq, finiteSeedDecode_encode] using hd |
| · intro h |
| simpa only [Function.comp_def, id_eq] using |
| h.comp measurable_id (measurable_finiteSeedEncode α) |
|
|
| section Reindex |
|
|
| variable {n bits bits' : ℕ} (A : OnlineRule n bits) (f : Seed bits' → Seed bits) |
|
|
| @[simp] theorem reindexOnlineSeed_decisionAt (r : Seed bits') (s v : Weights n) |
| (π : ArrivalOrder n) (k : Fin n) : |
| decisionAt (reindexOnlineSeed A f) r s v π k = decisionAt A (f r) s v π k := rfl |
| |
| @[simp] theorem reindexOnlineSeed_acceptedThrough (r : Seed bits') (s v : Weights n) |
| (π : ArrivalOrder n) (t : ℕ) : |
| acceptedThrough (reindexOnlineSeed A f) r s v π t = |
| acceptedThrough A (f r) s v π t := rfl |
|
|
| @[simp] theorem reindexOnlineSeed_reward (r : Seed bits') (s v : Weights n) |
| (π : ArrivalOrder n) : |
| reward (reindexOnlineSeed A f) r s v π = reward A (f r) s v π := rfl |
| |
| /-- Total decoding also preserves feasibility on encodings outside the support. -/ |
| theorem reindexOnlineSeed_feasible (M : Matroid (Fin n)) (hA : Feasible M A) : |
| Feasible M (reindexOnlineSeed A f) := by |
| intro r s v π t hs hv |
| exact hA (f r) s v π t hs hv |
| |
| variable (H : HiddenRule n bits) |
| |
| @[simp] theorem reindexHiddenSeed_acceptedThrough (r : Seed bits') (w : Weights n) |
| (π : ArrivalOrder n) (t : ℕ) : |
| hiddenAcceptedThrough (reindexHiddenSeed H f) r w π t = |
| hiddenAcceptedThrough H (f r) w π t := rfl |
|
|
| @[simp] theorem reindexHiddenSeed_reward (r : Seed bits') (w : Weights n) |
| (π : ArrivalOrder n) : |
| hiddenReward (reindexHiddenSeed H f) r w π = hiddenReward H (f r) w π := rfl |
| |
| @[simp] theorem reindexHiddenSeed_worstReward (r : Seed bits') (w : Weights n) : |
| hiddenWorstReward (reindexHiddenSeed H f) w r = hiddenWorstReward H w (f r) := rfl |
|
|
| /-- No independence of the adversary from the reindexed seed is used. -/ |
| theorem reindexHiddenSeed_integral (ν' : Measure (Seed bits')) |
| (ν : Measure (Seed bits)) (hlaw : ν'.map f = ν) (w : Weights n) : |
| (∫ r, hiddenWorstReward (reindexHiddenSeed H f) w r ∂ν') = |
| ∫ r, hiddenWorstReward H w r ∂ν := by |
| rw [← hlaw] |
| exact (integral_map_of_stronglyMeasurable (μ := ν') |
| (measurable_of_finite f) |
| (measurable_of_finite (hiddenWorstReward H w)).stronglyMeasurable).symm |
| |
| end Reindex |
| |
| end MatroidProphet |
| |
| end OAI |
| |
