| import OAI.Probability.FreeIsing.Ising.GaussianPosterior |
| import OAI.Probability.FreeIsing.Posterior.GaussianTesting |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal |
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| namespace Problem367 |
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| local instance : Nonempty IsingSpin := ⟨.neg⟩ |
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| 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 |
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| 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 |
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| end Problem367 |
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| end |
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| end OAI |
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