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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib
import OAI.Probability.LogConcave.Sampling.PrimitiveExpectedField
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open MeasureTheory MeasureTheory.Measure Function
open scoped Classical BigOperators RealInnerProductSpace NNReal
lemma gibbs_openPos {d : ℕ} {H : Point d → ℝ} (hH : Continuous H)
(hi : Integrable (fun z => Real.exp (-H z))) : IsOpenPosMeasure (gibbs H) := by
have hd : Measurable (gibbsDensity H) := (Real.continuous_exp.comp hH.neg).measurable.ennreal_ofReal
have hp : ∀ᵐ y : Point d ∂volume, gibbsDensity H y≠0 := Filter.Eventually.of_forall
(fun y => (ENNReal.ofReal_pos.mpr (Real.exp_pos _)).ne')
let : IsOpenPosMeasure (volume.withDensity (gibbsDensity H)) :=
(withDensity_absolutelyContinuous' hd.aemeasurable hp).isOpenPosMeasure
exact isOpenPosMeasure_smul (volume.withDensity (gibbsDensity H))
(ENNReal.inv_ne_zero.mpr (partition_ne_top_of_integrable hi))
lemma directional_adjoint_basis {d : ℕ} (A : Point d →L[ℝ] Point d)
(u : Point d) (φ : Point d → ℝ) (y : Point d) :
directional (A.adjoint u) φ y=∑i : Fin d,
inner ℝ u (A (EuclideanSpace.basisFun (Fin d) ℝ i))*
directional (EuclideanSpace.basisFun (Fin d) ℝ i) φ y := by
rw [directional_basis_sum]
apply Finset.sum_congr rfl
intro i _
rw [OrthonormalBasis.repr_apply_apply,ContinuousLinearMap.adjoint_inner_right,real_inner_comm]
theorem stein_weak_to_strong {d : ℕ} {H : Point d → ℝ}
(hH : ContDiff ℝ 1 H) (hi : Integrable (fun z => Real.exp (-H z)))
{K : Point d → (Point d →L[ℝ] Point d)} (hK : ContDiff ℝ 1 K)
{m : Point d → Point d} (hm : Continuous m)
(hw : ∀u φ,ContDiff ℝ 1 φ → HasCompactSupport φ →
(∫y,inner ℝ u (m y)*φ y ∂gibbs H)=∫y,directional ((K y).adjoint u) φ y ∂gibbs H)
(u : Point d) :
tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ)
(fun i y => inner ℝ u (K y (EuclideanSpace.basisFun (Fin d) ℝ i)))=
fun y => inner ℝ u (m y) := by
let b := EuclideanSpace.basisFun (Fin d) ℝ
let V := fun (i : Fin d) y => inner ℝ u (K y (b i))
have hV (i : Fin d) : ContDiff ℝ 1 (V i) :=
contDiff_const.inner ℝ (hK.clm_apply contDiff_const)
have hd : Continuous (tensorAdjoint H b V) := by
apply continuous_finsetSum
intro i _
exact (((hV i).continuous_fderiv (by norm_num)).clm_apply continuous_const).neg.add
(((hH.continuous_fderiv (by norm_num)).clm_apply continuous_const).mul (hV i).continuous)
have hc : Continuous (fun y => inner ℝ u (m y)) := continuous_const.inner hm
let : IsProbabilityMeasure (gibbs H) := probability_gibbs_of_partition
(partition_pos_of_continuous hH.continuous).ne' (partition_ne_top_of_integrable hi)
let : IsOpenPosMeasure (gibbs H) := gibbs_openPos hH.continuous hi
apply Measure.eq_of_ae_eq (μ:=gibbs H) _ hd hc
apply ae_eq_of_integral_contDiff_smul_eq (μ:=gibbs H) hd.locallyIntegrable hc.locallyIntegrable
intro φ hφ hφc
have hφ1 : ContDiff ℝ 1 φ := hφ.of_le (by simp)
have hp := tensorAdjoint_compact_pairing hH hi hV hφ1 hφc b
have ht := hw u φ hφ1 hφc
simp only [smul_eq_mul]
simp_rw [mul_comm (φ _)]; rw [hp,ht]
apply integral_congr_ae
filter_upwards [] with y
exact (directional_adjoint_basis (K y) u φ y).symm
theorem centeringKernel_strong {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) (hlam : 0<lam)
(hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) (u : Point d) :
tensorAdjoint (interpolationPotential F x r T) (EuclideanSpace.basisFun (Fin d) ℝ)
(fun i y => inner ℝ u (centeringKernel hF x hr hl hT0 hT1 y (EuclideanSpace.basisFun (Fin d) ℝ i)))=
fun y => inner ℝ u (conditionalFieldMean F x r T y-primitiveExpectedField F x r) := by
have hs := interpolationPotential_smooth hF x hr.le hl hT0 hT1
have hi := (interpolationPotential_lowerTail hF x hr.le (by linarith only [hl]) hT0 hT1).integrable_exp hs.continuous
apply stein_weak_to_strong (hs.of_le (by simp)) hi
((centeringKernel_smooth hF x hr hlam hl hT0 hT1).of_le (by simp))
((conditionalFieldMean_smooth hF x hr.le hl hT0 hT1).continuous.sub continuous_const) _ u
intro u φ hφ hφc
have hh := centeringKernel_weak hF x hr hlam hl hT0 hT1 u hφ hφc
rw [interpolationLaw_eq_gibbs hF x hr.le (by linarith only [hl])
(by nlinarith [probability_time hT0 hT1])] at hh
exact hh
end LogConcaveSampling
end
end
end
end OAI
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