| import Mathlib |
| import OAI.Probability.SKGap.Localization.CoordinateDeriv |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set |
| open scoped NNReal ENNReal |
| open Filter |
| open scoped Topology |
| namespace SKGap |
| 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 SKGap |
|
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| end |
|
|
| end OAI |
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|
