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a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.FiberDensity
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
lemma piGaussianConvolution_withDensity {m : ℕ} (F : (Fin m → ℝ) → ℝ≥0∞)
(hF : Measurable F) (δ : ℝ) (y : Fin m → ℝ) :
PiGaussianConvolution (volume.withDensity F) δ y =
(∫⁻ x, (∏ j, gaussianPDF 0 ⟨δ^2,sq_nonneg δ⟩ (y j-x j)) * F x).toReal := by
unfold PiGaussianConvolution
rw [integral_eq_lintegral_of_nonneg_ae
(Eventually.of_forall (fun x => Finset.prod_nonneg (fun j _ => gaussianPDFReal_nonneg _ _ _)))
((show Continuous (fun x : Fin m → ℝ => ∏ j,
gaussianPDFReal 0 ⟨δ^2,sq_nonneg δ⟩ (y j-x j)) from
by unfold gaussianPDFReal; fun_prop).aestronglyMeasurable),
lintegral_withDensity_eq_lintegral_mul _ hF (by fun_prop)]
congr 1
apply lintegral_congr
intro x
simp only [Pi.mul_apply, gaussianPDF]
rw [ENNReal.ofReal_prod_of_nonneg (fun j _ =>
gaussianPDFReal_nonneg 0 ⟨δ^2,sq_nonneg δ⟩ (y j-x j)), mul_comm]
lemma ae_tendsto_fiberGaussian {Ω : Type*} [MeasurableSpace Ω] (γ : Measure Ω)
[SigmaFinite γ] {m : ℕ} (hm : 1 ≤ m)
(F : Ω × (Fin m → ℝ) → ℝ≥0∞) (hF : Measurable F)
(hfin : (∫⁻ z, F z ∂γ.prod volume) ≠ ∞) :
∀ᵐ z ∂γ.prod volume, Tendsto
(fun i : ℕ => (fiberSmoothing volume F
(fun y : Fin m → ℝ => ∏ j, gaussianPDF 0
⟨(1/(i:ℝ))^2,sq_nonneg _⟩ (y j)) z).toReal)
atTop (𝓝 (F z).toReal) := by
have hmeas (i : ℕ) : Measurable (fun z => (fiberSmoothing volume F
(fun y : Fin m → ℝ => ∏ j, gaussianPDF 0
⟨(1/(i:ℝ))^2,sq_nonneg _⟩ (y j)) z).toReal) :=
(measurable_fiberSmoothing volume hF (by fun_prop)).ennreal_toReal
rw [Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hmeas hF.ennreal_toReal)]
have hinner : (∫⁻ A, ∫⁻ y : Fin m → ℝ, F (A,y) ∂volume ∂γ) ≠ ∞ := by
rwa [← lintegral_prod _ hF.aemeasurable]
filter_upwards [ae_lt_top (Measurable.lintegral_prod_right
(show Measurable (fun z : Ω × (Fin m → ℝ) => F (z.1,z.2)) from hF)) hinner] with A hA
let ν : Measure (Fin m → ℝ) := volume.withDensity (fun y => F (A,y))
have hFA : Measurable (fun y : Fin m → ℝ => F (A,y)) :=
hF.comp (measurable_const.prodMk measurable_id)
let : IsFiniteMeasure ν := isFiniteMeasure_withDensity hA.ne
have hac : ν ≪ volume := withDensity_absolutelyContinuous _ _
have hRN := Measure.rnDeriv_withDensity volume hFA
filter_upwards [ae_tendsto_piGaussianConvolution hm ν hac, hRN] with y hy hyRN
rw [hyRN] at hy
convert hy.comp reciprocal_nat_tendsto using 1
ext i
exact (piGaussianConvolution_withDensity (fun y => F (A,y)) hFA _ y).symm
lemma pointwiseDensity_ae_eq {d m : ℕ} (hm : 1 ≤ m) (μ : Measure (Space d))
[IsFiniteMeasure μ] (F : ((Fin m → Space d) × (Fin m → ℝ)) → ℝ)
(hFm : Measurable F) (hF0 : ∀ x, 0 ≤ F x)
(hden : (MatrixLabelReference d m).withDensity (fun x => ENNReal.ofReal (F x)) =
ProjectionLaw μ m) :
PointwiseDensity μ m =ᵐ[MatrixLabelReference d m] F := by
have hfin : (∫⁻ z, ENNReal.ofReal (F z) ∂MatrixLabelReference d m) ≠ ∞ := by
have he := congrArg (fun ν => ν Set.univ) hden
rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at he
rw [he]
let : IsFiniteMeasure (ProjectionLaw μ m) := by unfold ProjectionLaw; infer_instance
exact measure_ne_top _ _
have ht := ae_tendsto_fiberGaussian
(Measure.pi (fun _ : Fin m => stdGaussian (Space d))) hm
(fun x => ENNReal.ofReal (F x)) hFm.ennreal_ofReal hfin
have he (i : ℕ) := projectionSmoothing_ae_eq_fiberSmoothing μ m
(fun x => ENNReal.ofReal (F x)) hFm.ennreal_ofReal hden
(gaussianPDF 0 ⟨(1/(i:ℝ))^2,sq_nonneg _⟩) (measurable_gaussianPDF _ _)
filter_upwards [ht, ae_all_iff.mpr he] with x hx hxe
apply PointwiseDensity_eq_of_tendsto
rw [ENNReal.toReal_ofReal (hF0 x)] at hx
convert hx using 1
ext i
rw [GaussianRegularization_eq_toReal μ m _, hxe i]
end MemoryPrecision
end
end OAI
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