| import OAI.Probability.FreeIsing.Cavity.ArtanhFusion |
| import OAI.Probability.FreeIsing.Estimates.SelectedComponentFusion |
| import OAI.Probability.FreeIsing.Posterior.ConePosteriorRegression |
| import OAI.Probability.FreeIsing.Trees.SelectedPathLocality |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367 |
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| open PathSideBranches |
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| theorem HasFreeIsingLaw.selected_gaussian_outside_cone_fusion |
| {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 : Set (TreeVertex d)) (hS : S ⊆ (cone r)ᶜ) (hp : parentVertex r ∈ S) : |
| let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z |
| selectedPosterior P F (Subtype.val : (S ∪ cone r : Set (TreeVertex d)) → TreeVertex d) (parentVertex r) =ᵐ[P] |
| fun z => binaryFusion |
| (selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z) |
| (Real.tanh beta * selectedPosterior P F |
| (Subtype.val : cone r → TreeVertex d) r z) := by |
| let := hμ.isProbabilityMeasure |
| let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z |
| let X := fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| fun i : S => F (z.1 i.1) (z.2 i.1) |
| let Y := fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| fun i : cone r => F (z.1 i.1) (z.2 i.1) |
| have hF : Measurable (Function.uncurry F) := |
| ((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul |
| (t : ℝ)).add measurable_snd |
| have hX : Measurable X := by |
| apply Measurable.of_eval |
| intro i |
| exact hF.comp (show Measurable (fun z : |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| (z.1 i.1, z.2 i.1)) from by fun_prop) |
| have hY : Measurable Y := by |
| apply Measurable.of_eval |
| intro i |
| exact hF.comp (show Measurable (fun z : |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| (z.1 i.1, z.2 i.1)) from by fun_prop) |
| have hXout : Measurable[noisyRegionSigma (A := ℝ) (cone r)ᶜ] X := |
| (comap_measurable X).mono |
| ((selectedObservationField_le_noisyRegionSigma hF S).trans |
| (noisyRegionSigma_mono hS)) le_rfl |
| have hYin : Measurable[noisyRegionSigma (A := ℝ) (cone r)] Y := |
| (comap_measurable Y).mono |
| (selectedObservationField_le_noisyRegionSigma hF (cone r)) le_rfl |
| have hInd := hμ.condIndepFun_noisy_cone_observables |
| (Real.neg_one_lt_tanh beta).le (Real.tanh_lt_one beta).le |
| (gaussianReal 0 t) r hr X Y hXout hYin |
| have hreg : selectedPosterior P F (Subtype.val : cone r → TreeVertex d) |
| (parentVertex r) =ᵐ[P] fun z => Real.tanh beta * |
| selectedPosterior P F (Subtype.val : cone r → TreeVertex d) r z := |
| hμ.selectedPosterior_parent_cone |
| (Real.neg_one_lt_tanh beta).le (Real.tanh_lt_one beta).le |
| (gaussianReal 0 t) hF r hr |
| have hL : ∀ᵐ z ∂P, |
| |selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z| < 1 := by |
| filter_upwards [hμ.selected_gaussian_posterior_field_bound hbeta t |
| (Subtype.val : S → TreeVertex d) (parentVertex r) ⟨⟨_, hp⟩, rfl⟩] with z hz |
| exact abs_lt.mpr hz.1 |
| have hQ : ∀ᵐ z ∂P, |
| |selectedPosterior P F (Subtype.val : cone r → TreeVertex d) r z| < 1 := by |
| filter_upwards [hμ.selected_gaussian_posterior_field_bound hbeta t |
| (Subtype.val : cone r → TreeVertex d) r ⟨⟨r, List.prefix_refl _⟩, rfl⟩] with z hz |
| exact abs_lt.mpr hz.1 |
| have hR : ∀ᵐ z ∂P, |
| |selectedPosterior P F (Subtype.val : cone r → TreeVertex d) (parentVertex r) z| < 1 := by |
| filter_upwards [hreg, hQ] with z hz hq |
| rw [hz, abs_mul] |
| exact (mul_le_mul_of_nonneg_left hq.le (abs_nonneg (Real.tanh beta))).trans_lt |
| (by simpa using Real.abs_tanh_lt_one beta) |
| have h := condExp_spin_comap_sup_eq_binaryFusion |
| ((measurable_pi_apply (parentVertex r)).comp measurable_fst) hX hY |
| (hμ.planted_spin_event_mass (fun _ => gaussianReal 0 t) _) hInd hL hR |
| dsimp only |
| unfold selectedPosterior |
| rw [selectedObservationField_region_union] |
| filter_upwards [h, hreg] with z hz hregz |
| exact hz.trans (congrArg (binaryFusion |
| (selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z)) hregz) |
|
|
| theorem HasFreeIsingLaw.selected_gaussian_outside_cone_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 : Set (TreeVertex d)) (hS : S ⊆ (cone r)ᶜ) (hp : parentVertex r ∈ S) : |
| 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 ∪ cone r : Set (TreeVertex d)) → TreeVertex d) (parentVertex r) z)) =ᵐ[P] |
| fun z => Real.artanh (selectedPosterior P F |
| (Subtype.val : S → TreeVertex d) (parentVertex r) z) + |
| statisticMessage (Real.tanh beta) (selectedPosterior P F |
| (Subtype.val : cone r → TreeVertex d) r z) := by |
| have h := hμ.selected_gaussian_outside_cone_fusion hbeta t r hr S hS hp |
| have hL := hμ.selected_gaussian_posterior_field_bound hbeta t |
| (Subtype.val : S → TreeVertex d) (parentVertex r) ⟨⟨_, hp⟩, rfl⟩ |
| have hQ := hμ.selected_gaussian_posterior_field_bound hbeta t |
| (Subtype.val : cone r → TreeVertex d) r ⟨⟨r, List.prefix_refl _⟩, rfl⟩ |
| filter_upwards [h, hL, hQ] with z hz hl hq |
| rw [hz] |
| exact artanh_binaryFusion (abs_lt.mpr hl.1) |
| (binaryFusion_abs_mul_lt_one (Real.abs_tanh_lt_one beta) (abs_lt.mpr hq.1)) |
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| end Problem367 |
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| end |
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| end OAI |
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