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import OAI.Probability.LogConcave.Sampling.ConditionalFieldMean
import OAI.Probability.LogConcave.JetEstimates.Extension
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
open scoped RealInnerProductSpace
def conditionalSlotEmbedding {ι : Type*} (d : ℕ) (s : ι → Bool) :
(ι → Fin d) ↪ (ι → Fin d ⊕ Fin d) where
toFun a i := if s i then Sum.inr (a i) else Sum.inl (a i)
inj' := by
intro a c h
funext i
have hi := congrFun h i
cases hs : s i <;> simpa [hs] using hi
theorem conditional_cumulant_energy {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) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (y : Point d)
{ι ϑ : Type*} [Fintype ι] [DecidableEq ι] [Fintype ϑ] [DecidableEq ϑ]
(s : ι → Bool) (t : ϑ → Bool) (l : List ι) (k : List ϑ)
(hlN : l.Nodup) (hlall : l.toFinset=Finset.univ)
(hkN : k.Nodup) (hkall : k.toFinset=Finset.univ) (hlne : l≠[]) (hkne : k≠[]) :
let μ := jointConditionalLaw F x r ρ y ((lam:ℝ)*r)
let b := EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ
TensorEnergy.Bound (fun (a : ι → Fin d) (c : ϑ → Fin d) =>
JetCalculus.mixedJet (fun i => b (conditionalSlotEmbedding d s a i))
(fun j => b (conditionalSlotEmbedding d t c j)) l k
(fun z => Real.log (Appell.laplace μ (fun _ => (1:ℝ)) (z.1+z.2))))
(((Fintype.card ι+Fintype.card ϑ+1:ℕ):ℝ)^(2*(Fintype.card ι+Fintype.card ϑ+1)) *
Appell.splitConstant (2*Real.pi^2*(1-ρ^2)) (Fintype.card ι) (Fintype.card ϑ)) := by
obtain ⟨hp,hm,hP⟩ := jointConditionalLaw_properties hF x hr hlam hl hρ0 hρ1 y
let := hp
have hβ : 0 ≤ 2*Real.pi^2*(1-ρ^2) := by
obtain ⟨ha,_⟩ := probability_time hρ0 hρ1
positivity
exact (Appell.actual_cumulant_energy hm hβ hP
(EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ) l k hlN hlall hkN hkall hlne hkne).submatrix
(conditionalSlotEmbedding d s) (conditionalSlotEmbedding d t)
end LogConcaveSampling
namespace LogConcaveSampling
open MeasureTheory
open scoped RealInnerProductSpace
def conditionalU {d : ℕ} (F : Point d → ℝ) (x : Point d) (r ρ : ℝ)
(y u : Point d) (L : ℝ) {ι : Type*} (v : ι → Point d) (l : List ι) : ℝ :=
L/(1-ρ^2)^l.length * JetCalculus.jet (fun i => jointPosition d (v i)) l
(JetCalculus.dir (jointField d u)
(fun θ => Real.log (Appell.laplace (jointConditionalLaw F x r ρ y L)
(fun _ => (1:ℝ)) θ))) 0
lemma conditionalFieldMean_jet_U {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) (hρ0 : 0≤ρ) (hρ1 : ρ<1)
(u y : Point d) {ι : Type*} (v : ι → Point d) (l : List ι) :
JetCalculus.jet v l
(conditionalMeanScalar F x r ρ (fun z => inner ℝ u (primitiveField F x r z))) y=
ρ^l.length*conditionalU F x r ρ y u ((lam:ℝ)*r) v l := by
rw [conditionalFieldMean_jet hF x hr hlam hl hρ0 hρ1 u y v l]
simp only [conditionalU,div_pow]
ring
theorem conditional_jet_cumulant_energy {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) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (y : Point d)
{ι ϑ : Type*} [Fintype ι] [DecidableEq ι] [Fintype ϑ] [DecidableEq ϑ]
(s : ι → Bool) (t : ϑ → Bool) (l : List ι) (k : List ϑ)
(hlN : l.Nodup) (hlall : l.toFinset=Finset.univ)
(hkN : k.Nodup) (hkall : k.toFinset=Finset.univ) (hlne : l≠[]) (hkne : k≠[])
(o : List (ι ⊕ ϑ)) (ho : o.Perm (l.map Sum.inl++k.map Sum.inr)) :
let μ := jointConditionalLaw F x r ρ y ((lam:ℝ)*r)
let b := EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ
TensorEnergy.Bound (fun (a : ι → Fin d) (c : ϑ → Fin d) =>
JetCalculus.jet (Sum.elim (fun i => b (conditionalSlotEmbedding d s a i))
(fun j => b (conditionalSlotEmbedding d t c j))) o
(fun z => Real.log (Appell.laplace μ (fun _ => (1:ℝ)) z)) 0)
(((Fintype.card ι+Fintype.card ϑ+1:ℕ):ℝ)^(2*(Fintype.card ι+Fintype.card ϑ+1)) *
Appell.splitConstant (2*Real.pi^2*(1-ρ^2)) (Fintype.card ι) (Fintype.card ϑ)) := by
obtain ⟨hp,hm,hP⟩ := jointConditionalLaw_properties hF x hr hlam hl hρ0 hρ1 y
let := hp
have hβ : 0 ≤ 2*Real.pi^2*(1-ρ^2) := by
obtain ⟨ha,_⟩ := probability_time hρ0 hρ1
positivity
exact (Appell.actual_jet_cumulant_energy hm hβ hP
(EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ) l k hlN hlall hkN hkall hlne hkne o ho).submatrix
(conditionalSlotEmbedding d s) (conditionalSlotEmbedding d t)
lemma normalized_energy_factor {a : ℝ} (ha : a≠0) (B : ℝ) {j k : ℕ}
(hj : 0<j) (hk : 0<k) :
((a^(j+k-1))⁻¹)^2*(B*a)^(j+k)=B^(j+k)/(a^(j+k-2)) := by
obtain ⟨j,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hj)
obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hk)
have h1 : j.succ+k.succ-1=j+k+1 := by omega
have h2 : j.succ+k.succ-2=j+k := by omega
have h3 : j.succ+k.succ=(j+k)+2 := by omega
rw [h1,h2,h3,mul_pow]
simp only [pow_add,pow_one,pow_two]
field_simp
theorem conditional_normalized_tensor_energy {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) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (y : Point d)
{ι ϑ : Type*} [Fintype ι] [DecidableEq ι] [Fintype ϑ] [DecidableEq ϑ]
(s : ι → Bool) (t : ϑ → Bool) (l : List ι) (k : List ϑ)
(hlN : l.Nodup) (hlall : l.toFinset=Finset.univ)
(hkN : k.Nodup) (hkall : k.toFinset=Finset.univ) (hlne : l≠[]) (hkne : k≠[])
(o : List (ι ⊕ ϑ)) (ho : o.Perm (l.map Sum.inl++k.map Sum.inr)) :
let μ := jointConditionalLaw F x r ρ y ((lam:ℝ)*r)
let b := EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ
let n := Fintype.card ι+Fintype.card ϑ
TensorEnergy.Bound (fun (a : ι → Fin d) (c : ϑ → Fin d) =>
((1-ρ^2)^(n-1))⁻¹ * JetCalculus.jet (Sum.elim (fun i => b (conditionalSlotEmbedding d s a i))
(fun j => b (conditionalSlotEmbedding d t c j))) o
(fun z => Real.log (Appell.laplace μ (fun _ => (1:ℝ)) z)) 0)
(((n+1:ℕ):ℝ)^(2*(n+1))*(2*Real.pi^2)^n *
(((Fintype.card ι).factorial:ℝ)*((Fintype.card ϑ).factorial:ℝ))^2 /
(1-ρ^2)^(n-2)) := by
dsimp only
have hj : 0<Fintype.card ι := by
apply Fintype.card_pos_iff.mpr
obtain ⟨i,_⟩ := List.exists_mem_of_ne_nil l hlne
exact ⟨i⟩
have hk : 0<Fintype.card ϑ := by
apply Fintype.card_pos_iff.mpr
obtain ⟨i,_⟩ := List.exists_mem_of_ne_nil k hkne
exact ⟨i⟩
have hb := (conditional_jet_cumulant_energy hF x hr hlam hl hρ0 hρ1 y
s t l k hlN hlall hkN hkall hlne hkne o ho).smul
(((1-ρ^2)^(Fintype.card ι+Fintype.card ϑ-1))⁻¹)
convert! hb using 1
dsimp [Appell.splitConstant]
have he := normalized_energy_factor (probability_time hρ0 hρ1).1.ne' (2*Real.pi^2) hj hk
calc
_ = (((Fintype.card ι+Fintype.card ϑ+1:ℕ):ℝ)^(2*(Fintype.card ι+Fintype.card ϑ+1)) *
(((Fintype.card ι).factorial:ℝ)*((Fintype.card ϑ).factorial:ℝ))^2) *
((2*Real.pi^2)^(Fintype.card ι+Fintype.card ϑ)/(1-ρ^2)^(Fintype.card ι+Fintype.card ϑ-2)) := by ring
_ = _ := by rw [←he]; ring
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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