File size: 3,941 Bytes
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 | import OAI.Probability.FreeIsing.Trees.RootComponentAttenuation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ComponentSignalAttenuation
open PlantedSymmetry
theorem condExp_attenuation_le_critical_relabel_root_others
{d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1)
(c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (g : TreeAut d)
{B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
(G : NoisyRootOtherData ℝ c → B) (hG : Measurable G)
(shift : B → ℝ) (hs : Measurable shift) :
let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
let X := (G ∘ noisyRootOtherArray c) ∘ Prod.map (relabel g) (relabel g)
let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z)
(fun i => g.toEquiv.symm (componentIndexVertex (c := c) i))
(g.toEquiv.symm (RootedBroadcast.branchRoot c))
∀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
let := hμ.isProbabilityMeasure
let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) →
((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) :=
Prod.map (relabel g) (relabel g)
let X := G ∘ noisyRootOtherArray (A := ℝ) c
let f := fun v => responseAttenuation theta
(shift (X v) + statisticMessage theta (posteriorMean μ c t v)) ^ 2
have hX : Measurable X := hG.comp (measurable_noisyRootOtherArray c)
have hθsq : theta ^ 2 ≤ 1 := by nlinarith
have hfm : Measurable f :=
((continuous_responseAttenuation hθsq).measurable.comp
((hs.comp hX).add ((measurable_statisticMessage theta).comp
(measurable_posteriorMean μ c t)))).pow_const 2
have hfi : Integrable f P := by
apply (integrable_const (1 : ℝ)).mono' hfm.aestronglyMeasurable
filter_upwards [] with v
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hpos := responseAttenuation_pos hθsq
(shift (X v) + statisticMessage theta (posteriorMean μ c t v))
have hle := responseAttenuation_le_one hθsq
(shift (X v) + statisticMessage theta (posteriorMean μ c t v))
dsimp only [f]
nlinarith
have hmp : MeasurePreserving T P P :=
PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g
have hCE := condExp_comp_measurePreserving P P T hmp (mB.comap X) hX.comap_le hfi
have hbase := hmp.quasiMeasurePreserving.ae
(condExp_attenuation_le_critical_root_others hμ hθ0 hθ1 c hcritical ht G hG shift hs)
have hchannel : Measurable (Function.uncurry
(fun s z => (t : ℝ) * spinSign s + z)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
have hpost := selectedPosterior_relabel_ae hμ (gaussianReal 0 t) hchannel
(componentIndexVertex (c := c)) (RootedBroadcast.branchRoot c) g
have hfc : f ∘ T =ᵐ[P] fun v => responseAttenuation theta
(shift ((X ∘ T) v) + statisticMessage theta
(selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z)
(fun i => g.toEquiv.symm (componentIndexVertex (c := c) i))
(g.toEquiv.symm (RootedBroadcast.branchRoot c)) v)) ^ 2 := by
filter_upwards [hpost] with v hv
exact congrArg (fun q => responseAttenuation theta
(shift ((X ∘ T) v) + statisticMessage theta q) ^ 2) hv
have hcongr := condExp_congr_ae (m := mB.comap (X ∘ T)) hfc
rw [MeasurableSpace.comap_comp] at hCE
filter_upwards [hCE, hcongr, hbase] with w hw hc hb
exact hc.symm.trans_le (hw.trans_le hb)
end Problem367.ComponentSignalAttenuation
end
end OAI
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