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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 | import OAI.Probability.FreeIsing.Energy.RelabeledComponentAttenuation
import OAI.Probability.FreeIsing.Trees.NoisyPathLocality
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.PathSideBranches
theorem range_relabel_componentIndexVertex_eq_cone
{d : ℕ} (g : TreeAut d) (r : TreeVertex d) (c : Fin d)
(hgeo : ∀ u : TreeVertex d, u ∈ cone r ↔ [c] <+: (g.toEquiv u).1) :
Set.range (fun i => g.toEquiv.symm (componentIndexVertex (c := c) i)) =
Set.range (Subtype.val : cone r → TreeVertex d) := by
rw [Subtype.range_coe_subtype]
ext u
constructor
· rintro ⟨i, rfl⟩
apply (hgeo _).mpr
simp only [Equiv.apply_symm_apply]
change [c] <+: c :: i.2.1
simp
· intro hu
have hp := (hgeo u).mp hu
have hm : g.toEquiv u ∈ Set.range (componentIndexVertex (c := c)) := by
apply mem_range_componentIndexVertex.mpr
obtain ⟨w, hw⟩ := hp
exact ⟨w, by simpa using hw.symm⟩
obtain ⟨i, hi⟩ := hm
refine ⟨i, ?_⟩
change g.toEquiv.symm (componentIndexVertex i) = u
rw [hi, Equiv.symm_apply_apply]
end Problem367.PathSideBranches
namespace Problem367.ComponentSignalAttenuation
open PlantedSymmetry PathSideBranches
theorem condExp_attenuation_le_critical_cone_complement
{d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1)
(hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (r : TreeVertex d) (hr : r.1 ≠ [])
{B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
(G : (({u : TreeVertex d // u ∉ cone r} → IsingSpin) ×
({u : TreeVertex d // u ∉ cone r} → ℝ)) → B) (hG : Measurable G)
(shift : B → ℝ) (hs : Measurable shift) :
let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
let X := G ∘ noisyRegionArray (cone r)ᶜ
let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z)
(Subtype.val : cone r → TreeVertex d) r
∀ᵐ w ∂P,
(P[fun v => responseAttenuation theta
(shift (X v) + statisticMessage theta (Q v)) ^ 2 | mB.comap X]) w ≤
1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by
obtain ⟨g, c, hp, hc, hgeo⟩ := exists_reroot_cone r hr
have hroot : g.toEquiv.symm (RootedBroadcast.branchRoot c) = r := by
apply g.toEquiv.injective
rw [Equiv.apply_symm_apply, hc]
rfl
have hchannelField := selectedObservationField_eq_of_range_eq
(fun s z => (t : ℝ) * spinSign s + z)
(range_relabel_componentIndexVertex_eq_cone g r c hgeo)
have hX : ((G ∘ decodeConeComplement (A := ℝ) g r c hgeo) ∘ noisyRootOtherArray c) ∘
Prod.map (relabel g) (relabel g) = G ∘ noisyRegionArray (cone r)ᶜ := by
funext z
exact congrArg G (decodeConeComplement_relabel g r c hgeo z)
have h := condExp_attenuation_le_critical_relabel_root_others
hμ hθ0 hθ1 c hcritical ht g (G ∘ decodeConeComplement g r c hgeo)
(hG.comp (measurable_decodeConeComplement g r c hgeo)) shift hs
dsimp only at h ⊢
simpa only [hX, hroot, selectedPosterior, hchannelField] using h
theorem condExp_attenuation_le_critical_cone_of_measurable
{d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1)
(hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (r : TreeVertex d) (hr : r.1 ≠ [])
{B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
(X : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) → B)
(hX : Measurable[noisyRegionSigma (A := ℝ) (cone r)ᶜ] X)
(shift : B → ℝ) (hs : Measurable shift) :
let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z)
(Subtype.val : cone r → TreeVertex d) r
∀ᵐ w ∂P,
(P[fun v => responseAttenuation theta
(shift (X v) + statisticMessage theta (Q v)) ^ 2 | mB.comap X]) w ≤
1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by
obtain ⟨G, hG, rfl⟩ := hX.exists_eq_measurable_comp
exact condExp_attenuation_le_critical_cone_complement
hμ hθ0 hθ1 hcritical ht r hr G hG shift hs
end Problem367.ComponentSignalAttenuation
end
end OAI
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