File size: 3,182 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 | import OAI.Probability.FreeIsing.Cavity.ArraySmallLogLoss
import OAI.Probability.FreeIsing.Estimates.ThresholdArithmetic
import OAI.Probability.FreeIsing.Ising.CopyStationarity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingCopyStationarity
open IsingObservationInnovation IsingObservationFactor
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem integrable_fieldDifference_sq
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
Integrable (fun q => (IsingConditionalCopyBounds.fieldDifference beta v t q) ^ 2)
(copies beta μ) := by
exact (IsingConditionalCopyBounds.fieldDifference_secondMoment_bounds hμ hbeta
(ObservableDisintegration.compProd_kernel (observationLaw μ)
(measurable_innovationMap beta)) t v).1.integrable_sq
theorem integrable_sitePair_sub_sq
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
Integrable (fun p : CavityArray.SitePair d => (p.1 v - p.2 v) ^ 2)
((copies beta μ).map (sitePair beta t)) := by
have hm : Measurable (fun p : CavityArray.SitePair d => (p.1 v - p.2 v) ^ 2) :=
(((measurable_pi_apply v).comp measurable_fst).sub
((measurable_pi_apply v).comp measurable_snd)).pow_const 2
exact (integrable_map_measure hm.aestronglyMeasurable
(measurable_sitePair beta t).aemeasurable).mpr
(integrable_fieldDifference_sq hμ hbeta t v)
theorem integral_sitePair_cavity_sub_sq_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (o v w : TreeVertex d) :
let h0 := tanh_nonneg_of_beta_nonneg hbeta
let h1 := Real.tanh_lt_one beta
let P := (copies beta μ).map (sitePair beta t)
(∫ p, (CavityArray.field h0 h1 p.1 v w -
CavityArray.field h0 h1 p.2 v w) ^ 2 ∂P) ≤
(∫ p : CavityArray.SitePair d, (p.1 o - p.2 o) ^ 2 ∂P) /
(1 - Real.tanh beta) ^ 2 := by
exact CavityArray.integral_field_sub_sq_le
(tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(measurePreserving_sitePair_relabel hμ t) o v w
(integrable_sitePair_sub_sq hμ hbeta t o)
theorem integral_sitePair_log_cavity_sub_sq_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (o v w : TreeVertex d) :
let h0 := tanh_nonneg_of_beta_nonneg hbeta
let h1 := Real.tanh_lt_one beta
let P := (copies beta μ).map (sitePair beta t)
(∫ p, (Real.log (responseAttenuation (Real.tanh beta) (CavityArray.field h0 h1 p.1 v w)) -
Real.log (responseAttenuation (Real.tanh beta) (CavityArray.field h0 h1 p.2 v w))) ^ 2 ∂P) ≤
cavityLogVarianceCoefficient (Real.tanh beta) *
(∫ p : CavityArray.SitePair d, (p.1 o - p.2 o) ^ 2 ∂P) := by
exact CavityArray.integral_log_field_sub_sq_le
(tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(measurePreserving_sitePair_relabel hμ t) o v w
(integrable_sitePair_sub_sq hμ hbeta t o)
end Problem367.IsingCopyStationarity
end
end OAI
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