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ddc84c5 | 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 | 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
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