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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 | 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
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