| import Mathlib |
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| namespace OAI |
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| universe u_I u_K u_B u_A |
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| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
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| namespace Problem310.AdaptiveLabels |
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| / |
| 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 |
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| / |
| 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) |
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| / |
| 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] |
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| / |
| 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 |
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| end Problem310.AdaptiveLabels |
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| end OAI |
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