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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.Probability.FreeIsing.Estimates.InfiniteComponentCavity
import OAI.Probability.FreeIsing.Ising.CopySmallLogLoss
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
namespace Problem367.IsingCavityTailTransport
open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity
open IsingCopySpectral IsingCopySmallLogLoss PlantedSymmetry CavityFieldLimit
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
omit [IsProbabilityMeasure μ] in
theorem cavityObservable_identDistrib_relabel [_hprob : IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(g : TreeAut d) (t : ℝ≥0) (v w : TreeVertex d) :
IdentDistrib (cavityObservable hbeta t v w)
(cavityObservable hbeta t (g.toEquiv.symm v) (g.toEquiv.symm w))
(observationLaw μ) (observationLaw μ) := by
have hm := measurable_cavityObservable hbeta t v w
have hg := measurePreserving_observationLaw_relabel hμ g
have hmap : IdentDistrib (fun x => cavityObservable hbeta t v w (relabel g x))
(cavityObservable hbeta t v w) (observationLaw μ) (observationLaw μ) := by
refine ⟨(hm.comp hg.measurable).aemeasurable, hm.aemeasurable, ?_⟩
change (observationLaw μ).map ((cavityObservable hbeta t v w) ∘ relabel g) = _
rw [← Measure.map_map hm hg.measurable, hg.map_eq]
exact hmap.symm.trans (IdentDistrib.of_ae_eq (hm.comp hg.measurable).aemeasurable
(cavityObservable_relabel_ae hμ hbeta g t v w))
theorem cavityObservable_identDistrib_root_component
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (c : Fin d) :
IdentDistrib
(cavityObservable hbeta t ⟨[c], List.IsChain.singleton c⟩ (ConnectedHull.root d))
(fun p => Real.artanh (selectedPosterior
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (componentIndexVertex (c := c))
⟨[c], List.IsChain.singleton c⟩ p))
(observationLaw μ) (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by
have hobs := (IsingConditionalCopyBounds.hasLaw_observationLaw_eval μ t).identDistrib
(show HasLaw (observation t)
((μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map (observation t))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) from
⟨(measurable_observation t).aemeasurable, rfl⟩)
have hm := measurable_infiniteCavityField hbeta
⟨[c], List.IsChain.singleton c⟩ (ConnectedHull.root d)
have h := hobs.comp hm
change IdentDistrib
(cavityObservable hbeta t ⟨[c], List.IsChain.singleton c⟩ (ConnectedHull.root d))
(fun p => infiniteCavityField hbeta ⟨[c], List.IsChain.singleton c⟩
(ConnectedHull.root d) (observation t p))
(observationLaw μ) (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) at h
exact h.trans (IdentDistrib.of_ae_eq ((hm.comp (measurable_observation t)).aemeasurable)
(hμ.infiniteCavityField_eq_component hbeta t c))
theorem exists_cavityObservable_identDistrib_root_component
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {v w : TreeVertex d} (hvw : treeAdj v w) :
∃ c : Fin d, IdentDistrib (cavityObservable hbeta t v w)
(fun p => Real.artanh (selectedPosterior
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (componentIndexVertex (c := c))
⟨[c], List.IsChain.singleton c⟩ p))
(observationLaw μ) (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by
obtain ⟨g, hg⟩ := exists_treeAut_to_root w
have hgv : treeAdj (⟨[], List.IsChain.nil⟩ : TreeVertex d) (g.toEquiv v) := by
rw [← hg]
exact (g.map_adj w v).2 hvw.symm
obtain ⟨c, hc⟩ : ∃ c : Fin d, (g.toEquiv v).1 = [c] := by
rcases hgv with ⟨c, hc⟩ | ⟨c, hc⟩
· exact ⟨c, by simpa using hc⟩
· have hlen := congrArg List.length hc
simp at hlen
have hvc : g.toEquiv v = ⟨[c], List.IsChain.singleton c⟩ := Subtype.ext hc
have h := cavityObservable_identDistrib_relabel hμ hbeta g.symm t v w
change IdentDistrib (cavityObservable hbeta t v w)
(cavityObservable hbeta t (g.toEquiv v) (g.toEquiv w))
(observationLaw μ) (observationLaw μ) at h
rw [hvc, hg] at h
exact ⟨c, h.trans (cavityObservable_identDistrib_root_component hμ hbeta t c)⟩
theorem cavityObservable_tail_le_of_component_tails
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {v w : TreeVertex d} (hvw : treeAdj v w) {a : ℝ} {B : ℝ≥0∞}
(hTail : ∀ c : Fin d,
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) {p | a < |Real.artanh
(selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (componentIndexVertex (c := c))
⟨[c], List.IsChain.singleton c⟩ p)|} ≤ B) :
observationLaw μ {x | a < |cavityObservable hbeta t v w x|} ≤ B := by
obtain ⟨c, hc⟩ := exists_cavityObservable_identDistrib_root_component hμ hbeta t hvw
have heq := (hc.comp measurable_abs).measure_mem_eq
(measurableSet_Ioi : MeasurableSet (Set.Ioi a))
exact heq.trans_le (hTail c)
theorem logVariance_small_time_of_component_tails
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(o : TreeVertex d) {v w : TreeVertex d} (hvw : treeAdj v w)
{t T : ℝ≥0} (ht : 0 < (t : ℝ)) (htT : t ≤ T) {c : ℝ}
(hTail : ∀ i : Fin d,
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) {p | (t : ℝ) ^ (1 / 8 : ℝ) <
|Real.artanh (selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (componentIndexVertex (c := i))
⟨[i], List.IsChain.singleton i⟩ p)|} ≤
ENNReal.ofReal (Real.exp (-c * (Real.log (1 / (t : ℝ))) ^ 2))) :
logVariance hbeta μ v w t ≤
cavityLogVarianceCoefficient (Real.tanh beta) * (t : ℝ) ^ (1 / 4 : ℝ) *
siteVariance beta μ o t +
min (cavityLogVarianceCoefficient (Real.tanh beta) * siteVariance beta μ o t)
(cavityLogTailCoefficient (Real.tanh beta)
(2 * (T : ℝ) + 2 * (d : ℝ) * beta) *
Real.exp (-c * (Real.log (1 / (t : ℝ))) ^ 2)) :=
logVariance_small_time_of_tail hμ hbeta o v w ht htT
(cavityObservable_tail_le_of_component_tails hμ hbeta t hvw hTail)
end Problem367.IsingCavityTailTransport
end
end OAI
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