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caec230 | 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 | import Mathlib
namespace OAI
universe u_I u_K u_B u_A
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Problem310.AdaptiveLabels
/-- Distinct table addresses read independent labels, including when some
addresses in the table are unused. -/
theorem measure_injective_constraints
{I : Type u_I} {K : Type u_K} {B : Type u_B} [Fintype I] [Fintype K] [MeasurableSpace B]
(ν : Measure B) [IsProbabilityMeasure ν]
(addr : I → K) (hinj : Function.Injective addr)
(s : I → Set B) (hs : ∀ i, MeasurableSet (s i)) :
(Measure.pi (fun _ : K => ν)) {ω | ∀ i, ω (addr i) ∈ s i} =
∏ i, ν (s i) := by
classical
have hind : iIndepFun (fun k (ω : K → B) => ω k)
(Measure.pi (fun _ : K => ν)) :=
iIndepFun_pi (fun _ => measurable_id.aemeasurable)
have h := (hind.precomp hinj).measure_inter_preimage_eq_mul
Finset.univ (fun i _ => hs i)
simp only [Finset.mem_univ, Set.iInter_true] at h
rw [show {ω : K → B | ∀ i, ω (addr i) ∈ s i} =
(⋂ i, (fun ω : K → B => ω (addr i)) ⁻¹' s i) by ext ω; simp]
rw [h]
apply Finset.prod_congr rfl
intro i _
exact (measurePreserving_eval (fun _ : K => ν) (addr i)).measure_preimage
(hs i).nullMeasurableSet
/-- An address choice may depend arbitrarily on a separate selector outcome.
After fixing that outcome, injectivity is the only required terminal-table fact. -/
theorem measure_adaptive_constraints
{A : Type u_A} {I : Type u_I} {K : Type u_K} {B : Type u_B} [MeasurableSpace A] [Fintype I] [Fintype K]
[MeasurableSpace B] (μ : Measure A) (ν : Measure B)
[IsProbabilityMeasure ν]
(addr : A → I → K) (hinj : ∀ a, Function.Injective (addr a))
(s : A → I → Set B) (hs : ∀ a i, MeasurableSet (s a i))
(hmeas : MeasurableSet {z : A × (K → B) |
∀ i, z.2 (addr z.1 i) ∈ s z.1 i}) :
(μ.prod (Measure.pi (fun _ : K => ν)))
{z : A × (K → B) | ∀ i, z.2 (addr z.1 i) ∈ s z.1 i} =
∫⁻ a, ∏ i, ν (s a i) ∂μ := by
rw [Measure.prod_apply hmeas]
apply lintegral_congr
intro a
exact measure_injective_constraints ν (addr a) (hinj a) (s a) (hs a)
/-- The conditional failure probability of adaptively addressed Boolean tests.
Only active tests impose a zero label; distinct active addresses suffice here
because we assume the stronger, routing-friendly injectivity of all addresses. -/
theorem measure_adaptive_failures
{A : Type u_A} {I : Type u_I} {K : Type u_K} [MeasurableSpace A] [Fintype I] [Fintype K]
(μ : Measure A) (ν : Measure Bool) [IsProbabilityMeasure ν]
(addr : A → I → K) (hinj : ∀ a, Function.Injective (addr a))
(active : A → I → Prop) [∀ a i, Decidable (active a i)]
(hmeas : MeasurableSet {z : A × (K → Bool) |
∀ i, active z.1 i → z.2 (addr z.1 i) = false}) :
(μ.prod (Measure.pi (fun _ : K => ν)))
{z : A × (K → Bool) | ∀ i, active z.1 i → z.2 (addr z.1 i) = false} =
∫⁻ a, (ν {false}) ^ (Finset.univ.filter (active a)).card ∂μ := by
classical
let s : A → I → Set Bool := fun a i => if active a i then {false} else Set.univ
have hset : {z : A × (K → Bool) |
∀ i, active z.1 i → z.2 (addr z.1 i) = false} =
{z : A × (K → Bool) | ∀ i, z.2 (addr z.1 i) ∈ s z.1 i} := by
ext z
simp only [Set.mem_ofPred_eq]
apply forall_congr'
intro i
by_cases h : active z.1 i <;> simp [s, h]
rw [hset] at hmeas ⊢
rw [measure_adaptive_constraints μ ν addr hinj s
(fun _ _ => MeasurableSet.of_discrete) hmeas]
apply lintegral_congr
intro a
simp only [s, apply_ite, measure_univ]
simp [Finset.prod_ite]
/-- Looking up one terminal label at a selector-dependent address preserves its
marginal law. This is the pointwise mean-density identity for the routed set. -/
theorem measure_adaptive_label
{A : Type u_A} {K : Type u_K} {B : Type u_B} [MeasurableSpace A] [Fintype K]
[MeasurableSpace B] (μ : Measure A) (ν : Measure B)
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(addr : A → K) (s : Set B) (hs : MeasurableSet s)
(hmeas : MeasurableSet {z : A × (K → B) | z.2 (addr z.1) ∈ s}) :
(μ.prod (Measure.pi (fun _ : K => ν)))
{z : A × (K → B) | z.2 (addr z.1) ∈ s} = ν s := by
have h := measure_adaptive_constraints (I := Unit) μ ν
(fun a _ => addr a) (fun _ _ _ _ => Subsingleton.elim _ _)
(fun _ _ => s) (fun _ _ => hs) (by simpa using hmeas)
simpa using h
end Problem310.AdaptiveLabels
end OAI
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