| import OAI.Probability.FreeIsing.Estimates.InfiniteComponentCavity |
| import OAI.Probability.FreeIsing.Ising.CopySmallLogLoss |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal ENNReal |
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| namespace Problem367.IsingCavityTailTransport |
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| open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity |
| open IsingCopySpectral IsingCopySmallLogLoss PlantedSymmetry CavityFieldLimit |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| 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)) |
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| 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)) |
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| 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)⟩ |
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| 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) |
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| 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) |
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| end Problem367.IsingCavityTailTransport |
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| end |
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| end OAI |
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