| import OAI.Analysis.FiniteFactor.Coupling.Noncrossing |
| import OAI.Analysis.FiniteFactor.TailEstimate |
| import Mathlib.Probability.Kernel.CondDistrib |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal |
| noncomputable section |
| namespace CouplingParameters |
|
|
| /-- Identification with mathlib's regular conditional distribution. -/ |
| theorem vectorPosterior_eq_condDistrib {ι : Type*} [Countable ι] |
| (p : ι → CouplingParameters) (right : ι → Bool) : |
| condDistrib (vectorEndpoint (fun i => !right i)) (vectorEndpoint right) (vectorCoupling p) |
| =ᵐ[(vectorCoupling p).map (vectorEndpoint right)] vectorPosterior p right := |
| condDistrib_ae_eq_of_measure_eq_compProd_of_measurable |
| (vectorEndpoint_measurable right) (vectorEndpoint_measurable (fun i => !right i)) |
| (vectorPosterior_disintegration p right).symm |
|
|
| /-- A bound on the proved full-vector posterior is a conditional-expectation |
| bound on the original augmented probability space. -/ |
| theorem vector_condExp_event_le {ι : Type*} [Countable ι] |
| (p : ι → CouplingParameters) (right : ι → Bool) |
| {B : Set ((ι → ℝ) × (ι → ℝ))} (hB : MeasurableSet B) {δ : ℝ} |
| (hb : ∀ᵐ x ∂(vectorCoupling p).map (vectorEndpoint right), |
| (vectorPosterior p right x).real (Prod.mk x ⁻¹' B) ≤ δ) : |
| ∀ᵐ ω ∂vectorCoupling p, |
| (vectorCoupling p)[ |
| (fun ω => B.indicator (fun _ => (1 : ℝ)) |
| (vectorEndpoint right ω, vectorEndpoint (fun i => !right i) ω)) | |
| (inferInstance : MeasurableSpace (ι → ℝ)).comap (vectorEndpoint right)] ω ≤ δ := by |
| have hX := vectorEndpoint_measurable right |
| have hY := vectorEndpoint_measurable (fun i => !right i) |
| have hint : Integrable (B.indicator (fun _ => (1 : ℝ))) |
| ((vectorCoupling p).map (fun ω => (vectorEndpoint right ω, vectorEndpoint (fun i => !right i) ω))) := |
| (integrable_const _).indicator hB |
| have hc := condExp_prod_ae_eq_integral_condDistrib' hX hY.aemeasurable hint |
| have hk := ae_eq_comp hX.aemeasurable (vectorPosterior_eq_condDistrib p right) |
| have hb' := ae_of_ae_map hX.aemeasurable hb |
| filter_upwards [hc, hk, hb'] with ω hcω hkω hbω |
| simp only [Function.comp_apply] at hkω |
| rw [hcω, hkω] |
| have he : (fun y => B.indicator (fun _ => (1 : ℝ)) (vectorEndpoint right ω, y)) = |
| (Prod.mk (vectorEndpoint right ω) ⁻¹' B).indicator (fun _ => (1 : ℝ)) := rfl |
| rw [he, integral_indicator (measurable_prodMk_left hB), setIntegral_const] |
| simpa using hbω |
|
|
| /-- The countably infinite masked estimate, instantiated on the actual |
| augmented coupling using the verified posterior. No conditional probability |
| hypothesis remains to be supplied independently of the kernel. -/ |
| theorem vector_mask_tsum_integral_le {ι V : Type*} [Countable ι] [Countable V] |
| (p : ι → CouplingParameters) (right : ι → Bool) |
| (B : V → Set ((ι → ℝ) × (ι → ℝ))) (hB : ∀ v, MeasurableSet (B v)) |
| (f : V → (ι → Fin 6 × (ℝ × ℝ)) → ℝ) (δ : ℝ) |
| (hf : ∀ v, Integrable (f v) (vectorCoupling p)) |
| (hfm : ∀ v, StronglyMeasurable[(inferInstance : MeasurableSpace (ι → ℝ)).comap (vectorEndpoint right)] (f v)) |
| (hf0 : ∀ v ω, 0 ≤ f v ω) |
| (hsum : ∀ᵐ ω ∂vectorCoupling p, Summable (fun v => f v ω)) |
| (hint : Integrable (fun ω => ∑' v, f v ω) (vectorCoupling p)) |
| (hb : ∀ v, ∀ᵐ x ∂(vectorCoupling p).map (vectorEndpoint right), |
| (vectorPosterior p right x).real (Prod.mk x ⁻¹' B v) ≤ δ) : |
| (∫ ω, (∑' v, ((fun ω => (vectorEndpoint right ω, |
| vectorEndpoint (fun i => !right i) ω)) ⁻¹' B v).indicator (f v) ω) ∂vectorCoupling p) ≤ |
| δ * ∫ ω, (∑' v, f v ω) ∂vectorCoupling p := by |
| apply conditional_mask_tsum_integral_le (vectorEndpoint_measurable right).comap_le |
| hf hfm hf0 |
| (fun v => (hB v).preimage ((vectorEndpoint_measurable right).prodMk |
| (vectorEndpoint_measurable (fun i => !right i)))) hsum hint |
| intro v |
| exact vector_condExp_event_le p right (hB v) (hb v) |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
