| import OAI.Probability.FreeIsing.Trees.ConeSubsetFusion |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| namespace Problem367.PathSideBranches |
|
|
| theorem prefixFlip_mem_branch_iff {d : ℕ} (c : Fin d) (u : TreeVertex d) : |
| [c] <+: (prefixFlip c u).1 ↔ ¬ [c] <+: u.1 := by |
| rcases u with ⟨w, hw⟩ |
| cases w with |
| | nil => simp [prefixFlip, prefixFlipWord] |
| | cons a w => |
| by_cases h : c = a |
| · subst a |
| cases w with |
| | nil => simp [prefixFlip, prefixFlipWord] |
| | cons b w => |
| have hcb : c ≠ b := (List.isChain_cons_cons.mp hw).1 |
| simp [prefixFlip, prefixFlipWord, hcb] |
| · simp [prefixFlip, prefixFlipWord, h] |
|
|
| theorem exists_reroot_cone_complement {d : ℕ} (r : TreeVertex d) (hr : r.1 ≠ []) : |
| ∃ (g : TreeAut d) (c : Fin d), |
| g.toEquiv r = ConnectedHull.root d ∧ |
| g.toEquiv (parentVertex r) = ⟨[c], List.IsChain.singleton c⟩ ∧ |
| ∀ u : TreeVertex d, u ∉ cone r ↔ [c] <+: (g.toEquiv u).1 := by |
| obtain ⟨g, c, hp, hc, hgeo⟩ := exists_reroot_cone r hr |
| refine ⟨g.trans (prefixTreeAut c), c, ?_, ?_, ?_⟩ |
| · change prefixFlip c (g.toEquiv r) = _ |
| rw [hc] |
| apply Subtype.ext |
| simp [prefixFlip, prefixFlipWord, ConnectedHull.root] |
| · change prefixFlip c (g.toEquiv (parentVertex r)) = _ |
| rw [hp] |
| rfl |
| · intro u |
| change u ∉ cone r ↔ [c] <+: (prefixFlip c (g.toEquiv u)).1 |
| rw [prefixFlip_mem_branch_iff, hgeo] |
| end Problem367.PathSideBranches |
|
|
| namespace Problem367 |
| open PathSideBranches |
|
|
| theorem selectedPosterior_image_relabel_ae |
| {d : ℕ} {theta : ℝ} {A B : Type*} |
| [MeasurableSpace A] [MeasurableSpace B] |
| {μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) |
| (g : TreeAut d) (S : Set (TreeVertex d)) (r : TreeVertex d) : |
| let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)) |
| selectedPosterior P F (Subtype.val : g.toEquiv '' S → TreeVertex d) (g.toEquiv r) ∘ |
| Prod.map (relabel g) (relabel g) =ᵐ[P] |
| selectedPosterior P F (Subtype.val : S → TreeVertex d) r := by |
| have heq : Set.range (fun i : g.toEquiv '' S => g.toEquiv.symm i.1) = |
| Set.range (Subtype.val : S → TreeVertex d) := by |
| ext u |
| constructor |
| · rintro ⟨⟨v, ⟨x, hx, rfl⟩⟩, rfl⟩ |
| exact ⟨⟨x, hx⟩, by simp⟩ |
| · rintro ⟨x, rfl⟩ |
| exact ⟨⟨g.toEquiv x.1, ⟨x.1, x.2, rfl⟩⟩, by simp⟩ |
| have hfield := selectedObservationField_eq_of_range_eq F heq |
| have h := selectedPosterior_relabel_ae hμ ν hF |
| (Subtype.val : g.toEquiv '' S → TreeVertex d) (g.toEquiv r) g |
| simpa only [selectedPosterior, hfield, Equiv.symm_apply_apply] using h |
|
|
| theorem HasFreeIsingLaw.selectedPosterior_child_subset_complement |
| {d : ℕ} {theta : ℝ} {A B : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) |
| (r : TreeVertex d) (hr : r.1 ≠ []) |
| (T : Set (TreeVertex d)) (hT : T ⊆ (cone r)ᶜ) : |
| let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)) |
| selectedPosterior P F (Subtype.val : T → TreeVertex d) r =ᵐ[P] |
| fun z => theta * 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 hsub : g.toEquiv '' T ⊆ cone q := by |
| rintro u ⟨v, hv, rfl⟩ |
| exact (hgeo v).mp (hT hv) |
| have hbase := hμ.selectedPosterior_parent_subset_cone h0 h1 ν hF q hq |
| (g.toEquiv '' T) hsub |
| have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ ν g |
| have hb := hmp.quasiMeasurePreserving.ae_eq_comp hbase |
| have hleft := selectedPosterior_image_relabel_ae hμ ν hF g T r |
| have hright := selectedPosterior_image_relabel_ae hμ ν hF g T (parentVertex r) |
| rw [hr0] at hleft |
| rw [hpc] at hright |
| filter_upwards [hb, hleft, hright] with z hz hl hrz |
| rw [hpq] at hz |
| exact hl.symm.trans (hz.trans (congrArg (fun x : ℝ => theta * x) hrz)) |
|
|
| theorem HasFreeIsingLaw.selected_gaussian_subset_complement_field_difference |
| {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 U : Set (TreeVertex d)) (hS : S ⊆ cone r) |
| (hT : T ⊆ (cone r)ᶜ) (hU : U ⊆ (cone r)ᶜ) |
| (hp : r ∈ S) (ht : parentVertex r ∈ T) (hu : parentVertex r ∈ U) : |
| let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z |
| let field := fun (V : Set (TreeVertex d)) (u : TreeVertex d) z => |
| Real.artanh (selectedPosterior P F (Subtype.val : V → TreeVertex d) u z) |
| ∀ᵐ z ∂P, |
| |field (S ∪ T) r z - field (S ∪ U) r z| ≤ |
| Real.tanh beta * |field T (parentVertex r) z - field U (parentVertex r) z| ∧ |
| |field (S ∪ T) r z - field (S ∪ U) r z| ≤ 2 * beta := 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 hU' : g.toEquiv '' U ⊆ cone q := by |
| rintro u ⟨v, hv, rfl⟩ |
| exact (hgeo v).mp (hU 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 hu' : q ∈ g.toEquiv '' U := ⟨parentVertex r, hu, hpc⟩ |
| have hbase := hμ.selected_gaussian_subset_cone_field_difference hbeta t q hq |
| (g.toEquiv '' S) (g.toEquiv '' T) (g.toEquiv '' U) hS' hT' hU' hp' ht' hu' |
| have hmp := PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g |
| have hb := hmp.quasiMeasurePreserving.ae 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 hSU := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g (S ∪ U) r |
| have hTT := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g T (parentVertex r) |
| have hUU := selectedPosterior_image_relabel_ae hμ (gaussianReal 0 t) hF g U (parentVertex r) |
| filter_upwards [hb, hST, hSU, hTT, hUU] with z hz hst hsu htt huu |
| have hpc' : g.toEquiv (parentVertex r) = q := hpc |
| rw [hpq, ← hr0, ← hpc', ← Set.image_union, ← Set.image_union] at hz |
| simp only [Function.comp_apply] at hst hsu htt huu |
| dsimp only [F, PlantedSymmetry.gaussianNoiseLaw] at hst hsu htt huu hz ⊢ |
| simpa only [hst, hsu, htt, huu] using hz |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
