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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKRatio.Matrices.FiniteCoordinateDeriv
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
namespace SKRatioGaussian
variable {κ : Type*} [Fintype κ] [DecidableEq κ]
theorem gaussianFinite_IBP_error (i : κ)
{f d : (κ → ℝ) → ℝ} {L : ℝ≥0} {C D : ℝ} {s : Set (κ → ℝ)}
(hf : LipschitzWith L f) (hC : ∀ x, ‖f x‖ ≤ C)
(hd : Measurable d) (hD : ∀ x, ‖d x‖ ≤ D) (hs : MeasurableSet s)
(hder : ∀ x ∈ s, HasDerivAt (fun z => f (Function.update x i z)) (d x) (x i)) :
|(∫ x, x i*f x ∂Measure.pi (fun _ : κ => gaussianReal 0 1))-
∫ x, d x ∂Measure.pi (fun _ : κ => gaussianReal 0 1)| ≤
((L:ℝ)+D)*(Measure.pi (fun _ : κ => gaussianReal 0 1)).real sᶜ := by
let μ := Measure.pi (fun _ : κ => gaussianReal 0 1)
have hi : Integrable (finiteCoordinateDeriv i f) μ :=
Integrable.of_bound (measurable_finiteCoordinateDeriv i hf.continuous).aestronglyMeasurable
(L:ℝ) (ae_of_all _ (norm_finiteCoordinateDeriv_le i hf))
have hid : Integrable d μ := Integrable.of_bound hd.aestronglyMeasurable D (ae_of_all _ hD)
rw [gaussianFinite_lipschitz_IBP i hf hC,← integral_sub hi hid]
have he : (fun x => finiteCoordinateDeriv i f x-d x) =
sᶜ.indicator (fun x => finiteCoordinateDeriv i f x-d x) := by
funext x
by_cases hx : x ∈ s
· simp only [Set.indicator_of_notMem (show x ∉ sᶜ by simpa using hx),finiteCoordinateDeriv]
rw [(hder x hx).deriv,sub_self]
· rw [Set.indicator_of_mem hx]
rw [he,integral_indicator hs.compl]
exact norm_setIntegral_le_of_norm_le_const (measure_lt_top μ sᶜ) (fun x _ =>
(norm_sub_le _ _).trans (add_le_add (norm_finiteCoordinateDeriv_le i hf x) (hD x)))
end SKRatioGaussian
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
end
end
end
end
end OAI
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