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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 | import OAI.Probability.FreeIsing.Ising.GaussianPosterior
import OAI.Probability.FreeIsing.Posterior.GaussianTesting
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
namespace Problem367
local instance : Nonempty IsingSpin := ⟨.neg⟩
theorem HasFreeIsingLaw.restrict_real_eq_normalized_exp
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D)
(eta : D → IsingSpin) :
(μ.map D.restrict).real {eta} =
Real.exp (finiteIsingEnergy beta D eta) /
∑ xi, Real.exp (finiteIsingEnergy beta D xi) := by
rw [Measure.real, hμ.restrict_atom D hD eta, freeCylinderWeight_tanh_eq_exp]
rw [ENNReal.toReal_ofReal (by positivity)]
rw [show (∑ xi, Real.exp (finiteIsingEnergy beta D xi)) =
2 * (2 * Real.cosh beta) ^ (inducedEdges D).card from
finiteIsing_partitionFunction beta D hD]
rfl
theorem HasFreeIsingLaw.integral_standardGaussian_test_restrict_fieldMean
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D)
(t : ℝ) (ht : 0 ≤ t) (q : (D → IsingSpin) → ℝ)
(h : (D → ℝ) → ℝ) (hh : Measurable h) (C : ℝ)
(hbound : ∀ x, ‖h x‖ ≤ C) :
let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1))
let Y : (D → IsingSpin) × (D → ℝ) → D → ℝ :=
fun z v => t * spinSign (z.1 v) + Real.sqrt t * z.2 v
(∫ z, h (Y z) * q z.1 ∂ρ) =
∫ z, h (Y z) * FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun eta v => spinSign (eta v)) q (Y z) ∂ρ := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
exact finiteStandardGaussianPlanted_integral_test_fieldMean
(μ.map D.restrict) (finiteIsingEnergy beta D)
(hμ.restrict_real_eq_normalized_exp D hD) t ht
(fun eta v => spinSign (eta v)) (fun eta v => spinSign_sq (eta v))
q h hh C hbound
end Problem367
end
end OAI
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