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b317718 | 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 | import OAI.Probability.FreeIsing.Trees.ConeComplementFusion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PathSideBranches
theorem HasFreeIsingLaw.selected_gaussian_inside_subset_complement_field
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (r : TreeVertex d) (hr : r.1 ≠ [])
(S T : Set (TreeVertex d)) (hS : S ⊆ cone r) (hT : T ⊆ (cone r)ᶜ)
(hp : r ∈ S) (ht : parentVertex r ∈ T) :
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
(fun z => Real.artanh (selectedPosterior P F
(Subtype.val : (S ∪ T : Set (TreeVertex d)) → TreeVertex d) r z)) =ᵐ[P]
fun z => Real.artanh (selectedPosterior P F
(Subtype.val : S → TreeVertex d) r z) +
statisticMessage (Real.tanh beta) (selectedPosterior P F
(Subtype.val : T → TreeVertex d) (parentVertex r) z) := by
obtain ⟨g, c, hr0, hpc, hgeo⟩ := exists_reroot_cone_complement r hr
let q : TreeVertex d := ⟨[c], List.IsChain.singleton c⟩
have hq : q.1 ≠ [] := by simp [q]
have hpq : parentVertex q = ConnectedHull.root d := by rfl
have hS' : g.toEquiv '' S ⊆ (cone q)ᶜ := by
rintro u ⟨v, hv, rfl⟩ hbad
exact ((hgeo v).mpr hbad) (hS hv)
have hT' : g.toEquiv '' T ⊆ cone q := by
rintro u ⟨v, hv, rfl⟩
exact (hgeo v).mp (hT hv)
have hp' : parentVertex q ∈ g.toEquiv '' S := ⟨r, hp, hr0.trans hpq.symm⟩
have ht' : q ∈ g.toEquiv '' T := ⟨parentVertex r, ht, hpc⟩
have hbase := hμ.selected_gaussian_outside_subset_cone_field hbeta t q hq
(g.toEquiv '' S) (g.toEquiv '' T) hS' hT' hp' ht'
have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g
have hb := hmp.quasiMeasurePreserving.ae_eq_comp hbase
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
have hF : Measurable (Function.uncurry F) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
have hST := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g (S ∪ T) r
have hSS := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g S r
have hTT := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g T (parentVertex r)
filter_upwards [hb, hST, hSS, hTT] with z hz hst hss htt
have hpc' : g.toEquiv (parentVertex r) = q := hpc
rw [hpq, ← hr0, ← hpc', ← Set.image_union] at hz
simp only [Function.comp_apply] at hst hss htt
dsimp only [F, PlantedSymmetry.gaussianNoiseLaw] at hst hss htt hz ⊢
simpa only [Function.comp_apply, hst, hss, htt] using hz
end Problem367
end
end OAI
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