| import OAI.Probability.FreeIsing.Trees.ConeComplementFusion |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| namespace Problem367 |
| open PathSideBranches |
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| 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 |
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| end Problem367 |
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| end |
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| end OAI |
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