openAI-math-dataset / lean /OAI /Probability /LogConcave /Sampling /JointConditionalLawProperties.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.MaterialScalar | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open MeasureTheory ProbabilityTheory | |
| open scoped Topology Classical BigOperators NNReal RealInnerProductSpace | |
| lemma jointConditionalLaw_properties_sq {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} | |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) | |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ : ρ^2<1) (y : Point d) : | |
| IsProbabilityMeasure (jointConditionalLaw F x r ρ y ((lam:ℝ)*r)) ∧ | |
| Appell.HasExpMoments (jointConditionalLaw F x r ρ y ((lam:ℝ)*r)) := by | |
| let L : ℝ≥0 := lam*(NNReal.mk r hr.le) | |
| have hL : L≠0 := mul_ne_zero hlam.ne' (by intro h; exact hr.ne' (congrArg (fun t : ℝ≥0 => (t:ℝ)) h)) | |
| have hK := jointImage_lipschitz (primitiveField_lipschitz hF x hr.le) hL | |
| let := conditionalLaw_probability_sq hF x hr.le hl hρ y | |
| change IsProbabilityMeasure ((gibbs (conditionalPotential F x r ρ y)).map | |
| (jointImage (primitiveField F x r) (L:ℝ))) ∧ _ | |
| exact ⟨inferInstance, | |
| (conditionalLaw_hasExpMoments_sq hF x hr.le hl hρ y).map_lipschitz hK⟩ | |
| lemma conditionalU_normalized_jet_sq {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} | |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) | |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ : ρ^2<1) | |
| {ι : Type*} (v : ι → Point d) (l : List ι) (u y : Point d) : | |
| conditionalU F x r ρ y u ((lam:ℝ)*r) v l= | |
| ((1-ρ^2)^l.length)⁻¹*JetCalculus.jet v l | |
| (Appell.normalizedLaplace (gibbs (conditionalPotential F x r ρ y)) | |
| (fun z => inner ℝ u (primitiveField F x r z))) 0 := by | |
| obtain ⟨hp,hJ⟩ := jointConditionalLaw_properties_sq hF x hr hlam hl hρ y | |
| let := hp | |
| let := conditionalLaw_probability_sq hF x hr.le hl hρ y | |
| have hc : Continuous (fun z => inner ℝ u (primitiveField F x r z)) := | |
| continuous_const.inner (primitiveField_contDiff hF x r).continuous | |
| have hg := primitiveField_inner_growth hF x u hr.le | |
| have hC := Appell.smooth_normalizedLaplace (conditionalLaw_hasExpMoments_sq hF x hr.le hl hρ y) hc hg | |
| have hJC := Appell.smooth_normalizedLaplace hJ | |
| (show Continuous (fun z : JointPoint d => inner ℝ (jointField d u) z) by fun_prop) | |
| (Appell.HasGrowth.linear (innerSL ℝ (jointField d u))) | |
| have hh := congrFun (JetCalculus.jet_comp_affine hJC (jointPosition d) 0 v l) 0 | |
| simp only [add_zero,map_zero] at hh | |
| have hf : (fun z => Appell.normalizedLaplace (jointConditionalLaw F x r ρ y ((lam:ℝ)*r)) | |
| (fun z => inner ℝ (jointField d u) z) (jointPosition d z))=fun z => | |
| ((lam:ℝ)*r)⁻¹*Appell.normalizedLaplace (gibbs (conditionalPotential F x r ρ y)) | |
| (fun z => inner ℝ u (primitiveField F x r z)) z := by | |
| funext z | |
| exact joint_normalizedLaplace (primitiveField_contDiff hF x r).continuous _ _ z | |
| rw [hf,JetCalculus.jet_const_mul hC] at hh | |
| unfold conditionalU | |
| rw [Appell.dir_log_laplace hJ,←hh] | |
| dsimp only at hh | |
| have hL : (lam:ℝ)*r≠0 := by positivity | |
| field_simp [hL] | |
| end LogConcaveSampling | |
| end | |
| end | |
| end | |
| end OAI | |