openAI-math-dataset / lean /OAI /Probability /LogConcave /JetEstimates /JetNormalizedLaplaceMoments.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.JetEstimates.NormalizedLaplace | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling.Appell | |
| open MeasureTheory ProbabilityTheory | |
| open scoped Classical BigOperators RealInnerProductSpace | |
| variable {E : Type} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] | |
| variable {ι : Type*} [DecidableEq ι] | |
| lemma jet_normalizedLaplace_zero_moments {μ : Measure E} [IsProbabilityMeasure μ] | |
| (hμ : HasExpMoments μ) {h : E → ℝ} (hh : Continuous h) (hg : HasGrowth h) | |
| (v : ι → E) (l : List ι) (hl : l.Nodup) : | |
| JetCalculus.jet v l (normalizedLaplace μ h) 0= | |
| ∑s∈l.toFinset.powerset,inverseMoment (moments μ (fun i z => inner ℝ (v i) z)) s* | |
| ∫z,(∏i∈l.toFinset\s,inner ℝ (v i) z)*h z ∂μ := by | |
| rw [jet_normalizedLaplace_zero hμ hh hg v l hl] | |
| simp only [polynomial,Finset.sum_mul] | |
| rw [integral_finsetSum] | |
| · apply Finset.sum_congr rfl | |
| intro s _ | |
| rw [show (fun z => (inverseMoment (moments μ (fun i z => inner ℝ (v i) z)) s* | |
| ∏i∈l.toFinset\s,inner ℝ (v i) z)*h z)= | |
| fun z => inverseMoment (moments μ (fun i z => inner ℝ (v i) z)) s* | |
| ((∏i∈l.toFinset\s,inner ℝ (v i) z)*h z) by funext z; ring, | |
| integral_const_mul] | |
| · intro s _ | |
| exact ((HasGrowth.const _).mul ((HasGrowth.prod (l.toFinset\s) | |
| (fun i _ => HasGrowth.linear (innerSL ℝ (v i)))))).mul hg |>.integrable | |
| (((continuous_const.mul (continuous_finsetProd _ | |
| (fun i _ => continuous_const.inner continuous_id))).mul hh).aestronglyMeasurable) hμ.hasMoments | |
| end LogConcaveSampling.Appell | |
| end | |
| end | |
| end | |
| end OAI | |