| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.QuadraticExpectation |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology Classical BigOperators NNReal RealInnerProductSpace |
|
|
| namespace Appell |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| variable {ι : Type*} [DecidableEq ι] |
|
|
| lemma inverseMoment_contDiffAt {m : E → Finset ι → ℝ} {p : E} |
| (hm : ∀s,ContDiffAt ℝ (⊤:ℕ∞) (fun x => m x s) p) (s : Finset ι) : |
| ContDiffAt ℝ (⊤:ℕ∞) (fun x => inverseMoment (m x) s) p := by |
| induction s using Finset.strongInductionOn with |
| | _ s ih => |
| by_cases hs : s=∅ |
| · subst s |
| simp only [inverseMoment_empty] |
| exact contDiffAt_const |
| simp_rw [inverseMoment_nonempty _ hs] |
| apply ContDiffAt.neg |
| apply ContDiffAt.sum |
| intro t ht |
| exact (hm (s\t)).mul (ih t (Finset.ssubset_iff_subset_ne.mpr |
| ⟨Finset.mem_powerset.mp (Finset.mem_erase.mp ht).2,(Finset.mem_erase.mp ht).1⟩)) |
|
|
| end Appell |
|
|
| lemma conditionalMeanScalar_joint_smooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2<1) {h : Point d → ℝ} (hc : Continuous h) |
| (hg : Appell.HasGrowth h) (p : ℝ × Point d) (hp : p.1^2<1) : |
| ContDiffAt ℝ (⊤:ℕ∞) (fun q : ℝ × Point d => conditionalMeanScalar F x r q.1 h q.2) p := by |
| obtain ⟨m,hm,C,ht⟩ := hF.hasGaussianLowerTail x hr hl |
| have hp' : 0 < m +(conditionalParameter p).2 := by dsimp [conditionalParameter]; positivity |
| have hh := (contDiffAt_quadraticExpectation (hF.continuous_potential x r) |
| hc ht hg (conditionalParameter p) hp').comp p (conditionalParameter_smooth hp) |
| apply hh.congr_of_eventuallyEq |
| filter_upwards [(isOpen_lt (continuous_fst.pow 2) continuous_const).mem_nhds hp] with q hq |
| exact conditional_integral_quadratic hF x hr hl hq q.2 h |
|
|
| def conditionalPositionMoment {d : ℕ} {ι : Type*} [DecidableEq ι] |
| (F : Point d → ℝ) (x : Point d) (r : ℝ) (v : ι → Point d) (s : Finset ι) |
| (p : ℝ × Point d) : ℝ := |
| conditionalMeanScalar F x r p.1 (fun z => ∏i∈s,inner ℝ (v i) z) p.2 |
|
|
| def conditionalMixedMoment {d : ℕ} {ι : Type*} [DecidableEq ι] |
| (F : Point d → ℝ) (x : Point d) (r : ℝ) (u : Point d) |
| (v : ι → Point d) (s : Finset ι) (p : ℝ × Point d) : ℝ := |
| conditionalMeanScalar F x r p.1 |
| (fun z => (∏i∈s,inner ℝ (v i) z)*inner ℝ u (primitiveField F x r z)) p.2 |
|
|
| lemma conditionalPositionMoment_smooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2<1) {ι : Type*} [DecidableEq ι] |
| (v : ι → Point d) (s : Finset ι) (p : ℝ × Point d) (hp : p.1^2<1) : |
| ContDiffAt ℝ (⊤:ℕ∞) (conditionalPositionMoment F x r v s) p := |
| conditionalMeanScalar_joint_smooth hF x hr hl |
| (continuous_finsetProd _ (fun _ _ => continuous_const.inner continuous_id)) |
| (Appell.HasGrowth.prod s (fun i _ => Appell.HasGrowth.linear (innerSL ℝ (v i)))) p hp |
|
|
| lemma conditionalMixedMoment_smooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2<1) {ι : Type*} [DecidableEq ι] |
| (u : Point d) (v : ι → Point d) (s : Finset ι) (p : ℝ × Point d) (hp : p.1^2<1) : |
| ContDiffAt ℝ (⊤:ℕ∞) (conditionalMixedMoment F x r u v s) p := |
| conditionalMeanScalar_joint_smooth hF x hr hl |
| ((continuous_finsetProd _ (fun _ _ => continuous_const.inner continuous_id)).mul |
| (continuous_const.inner (primitiveField_contDiff hF x r).continuous)) |
| ((Appell.HasGrowth.prod s (fun i _ => Appell.HasGrowth.linear (innerSL ℝ (v i)))).mul |
| (primitiveField_inner_growth hF x u hr)) p hp |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
