| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.ProbabilityFlowIntegral |
| import OAI.Probability.LogConcave.Sampling.MeasureSmoothLipschitz |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory |
|
|
| theorem probabilityFlow_pushforward {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) : |
| (stdGaussian (Point d)).map (probabilityFlow hF x hr hl hT0.le hT1)= |
| interpolationLaw F x r T := by |
| let := probability_gibbs_of_partition |
| (partition_pos_of_continuous (hF.continuous_potential x r)).ne' |
| (partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith))) |
| have hc := probabilityFlow_continuous hF x hr hl hT0.le hT1 |
| let hprob : IsProbabilityMeasure (interpolationLaw F x r T) := by |
| unfold interpolationLaw |
| infer_instance |
| apply measure_eq_of_smooth_lipschitz |
| intro φ hφ ⟨L,hL⟩ ⟨B,hB⟩ |
| rw [integral_map hc.measurable.aemeasurable hφ.continuous.aestronglyMeasurable] |
| exact (probabilityFlow_integral hF x hr hl hT0 hT1 (hφ.differentiable (by norm_num)) hL hB).symm |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
