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dd14c26 | 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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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