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import OAI.Probability.LogConcave.Sampling.CoordinateProjection
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open MeasureTheory Set ProbabilityTheory
open scoped ENNReal NNReal
lemma productPotential_measurable {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : Measurable H) (hG : Measurable G) : Measurable (productPotential H G) :=
(hH.comp ((productPointEquiv d e).continuous.measurable.fst)).add
(hG.comp ((productPointEquiv d e).continuous.measurable.snd))
lemma productPotential_unnormalized {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : Measurable H) (hG : Measurable G) :
(volume.withDensity (gibbsDensity (productPotential H G))).map (productPointEquiv d e)=
(volume.withDensity (gibbsDensity H)).prod (volume.withDensity (gibbsDensity G)) := by
let E := (productPointEquiv d e).toHomeomorph.toMeasurableEquiv
have hd : Measurable (gibbsDensity (productPotential H G)) :=
(Real.measurable_exp.comp (productPotential_measurable hH hG).neg).ennreal_ofReal
have hHd : Measurable (gibbsDensity H) := (Real.measurable_exp.comp hH.neg).ennreal_ofReal
have hGd : Measurable (gibbsDensity G) := (Real.measurable_exp.comp hG.neg).ennreal_ofReal
rw [show (productPointEquiv d e : Point (d+e) → Point d × Point e)=E from rfl,
map_withDensity_of_measurePreserving E (productPointEquiv_volume d e) hd,
prod_withDensity hHd hGd]
congr 1
funext p
rw [productPotential_density]
change gibbsDensity H (E (E.symm p)).1*gibbsDensity G (E (E.symm p)).2=_
rw [E.apply_symm_apply]
lemma productPotential_partition {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : Measurable H) (hG : Measurable G) :
partition (productPotential H G)=partition H*partition G := by
have he := congrArg (fun μ : Measure (Point d × Point e) => μ univ)
(productPotential_unnormalized hH hG)
rw [Measure.map_apply (productPointEquiv d e).continuous.measurable MeasurableSet.univ,
preimage_univ] at he
have hp : ((volume.withDensity (gibbsDensity H)).prod (volume.withDensity (gibbsDensity G))) univ=
(volume.withDensity (gibbsDensity H)) univ*(volume.withDensity (gibbsDensity G)) univ := by
rw [← univ_prod_univ,Measure.prod_prod]
rw [hp] at he
simpa only [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,partition] using he
theorem productPotential_gibbs {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : Continuous H) (hG : Continuous G) :
(gibbs (productPotential H G)).map (productPointEquiv d e)=(gibbs H).prod (gibbs G) := by
unfold gibbs
rw [Measure.map_smul _ (productPointEquiv d e).continuous.measurable.aemeasurable,
productPotential_unnormalized hH.measurable hG.measurable,productPotential_partition hH.measurable hG.measurable,
ENNReal.mul_inv (Or.inl (partition_pos_of_continuous hH).ne') (Or.inr (partition_pos_of_continuous hG).ne'),Measure.prod_smul_left,Measure.prod_smul_right,smul_smul]
lemma gaussianPotential_gibbs (d : ℕ) :
gibbs (fun y : Point d => ‖y‖^2/2)=stdGaussian (Point d) := by
have he := congrArg (fun μ : Measure (Point d) => μ univ) (EulerDensity.standard_density d)
have hmul : EulerDensity.gaussianCoefficient d*partition (fun y : Point d => ‖y‖^2/2)=1 := by
simpa only [measure_univ,Measure.smul_apply,smul_eq_mul,
withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,partition,gibbsDensity,neg_div] using he.symm
have hi : (partition (fun y : Point d => ‖y‖^2/2))⁻¹=EulerDensity.gaussianCoefficient d := by
exact (ENNReal.eq_inv_of_mul_eq_one_left hmul).symm
rw [EulerDensity.standard_density,gibbs,hi]
congr 2
funext y
simp only [gibbsDensity,neg_div]
end LogConcaveSampling
end
end
end
end OAI
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