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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Probability.FreeIsing.Cavity.ArtanhFusion
import OAI.Probability.FreeIsing.Estimates.SelectedComponentFusion
import OAI.Probability.FreeIsing.Posterior.ConePosteriorRegression
import OAI.Probability.FreeIsing.Trees.SelectedPathLocality
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PathSideBranches
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))
end Problem367
end
end OAI
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