| import OAI.Probability.FreeIsing.Conditioning.FiniteBayesCondExp |
| import OAI.Probability.FreeIsing.Gaussian.BinaryFusion |
| import OAI.Probability.FreeIsing.Spins.Kernel |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal ProbabilityTheory |
|
|
| namespace Problem367 |
|
|
| local instance : MeasurableSingletonClass IsingSpin where |
| measurableSet_singleton _ := trivial |
|
|
| local instance : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| theorem binaryFusion_likelihood_nonneg {u v : ℝ} |
| (hu : |u| ≤ 1) (hv : |v| ≤ 1) (a : IsingSpin) : |
| 0 ≤ (1 + spinSign a * u) * (1 + spinSign a * v) := by |
| rcases abs_le.mp hu with ⟨hu₁, hu₂⟩ |
| rcases abs_le.mp hv with ⟨hv₁, hv₂⟩ |
| cases a <;> simp only [spinSign_neg, spinSign_pos, neg_one_mul, one_mul] <;> |
| apply mul_nonneg <;> linarith |
|
|
| theorem binaryFusion_likelihood_sum {u v : ℝ} |
| (hu : |u| ≤ 1) (hv : |v| ≤ 1) : |
| (∑ a : IsingSpin, ENNReal.ofReal (1 / 2 : ℝ) * |
| ENNReal.ofReal ((1 + spinSign a * u) * (1 + spinSign a * v))) = |
| ENNReal.ofReal (1 + u * v) := by |
| simp_rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1 / 2)] |
| rw [sum_isingSpin, ← ENNReal.ofReal_add |
| (mul_nonneg (by norm_num) (binaryFusion_likelihood_nonneg hu hv .neg)) |
| (mul_nonneg (by norm_num) (binaryFusion_likelihood_nonneg hu hv .pos))] |
| congr 1 |
| simp only [spinSign_neg, spinSign_pos] |
| ring |
|
|
| theorem posterior_spinMean_eq_binaryFusion |
| {X : Type*} [MeasurableSpace X] |
| (π : Measure IsingSpin) [IsProbabilityMeasure π] |
| (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (ν : Measure X) [SigmaFinite ν] |
| (κ : Kernel IsingSpin X) [IsFiniteKernel κ] |
| (u v : X → ℝ) (hu : Measurable u) (hv : Measurable v) |
| (hu₁ : ∀ᵐ x ∂ν, |u x| < 1) (hv₁ : ∀ᵐ x ∂ν, |v x| < 1) |
| (hκ : ∀ a, κ a = ν.withDensity |
| (fun x => ENNReal.ofReal ((1 + spinSign a * u x) * (1 + spinSign a * v x)))) : |
| ∀ᵐ x ∂κ ∘ₘ π, |
| (∫ a, spinSign a ∂(posterior κ π) x) = binaryFusion (u x) (v x) := by |
| let L : IsingSpin → X → ℝ≥0∞ := fun a x => |
| ENNReal.ofReal ((1 + spinSign a * u x) * (1 + spinSign a * v x)) |
| have hL (a : IsingSpin) : Measurable (L a) := |
| ((measurable_const.add (measurable_const.mul hu)).mul |
| (measurable_const.add (measurable_const.mul hv))).ennreal_ofReal |
| have hpos : ∀ᵐ x ∂ν, (∑ a, π {a} * L a x) ≠ 0 := by |
| filter_upwards [hu₁, hv₁] with x hux hvx |
| simp only [L, hπ] |
| rw [binaryFusion_likelihood_sum hux.le hvx.le] |
| exact ne_of_gt (ENNReal.ofReal_pos.mpr (binaryFusion_denominator_pos hux hvx)) |
| have hfinite : ∀ᵐ x ∂ν, (∑ a, π {a} * L a x) ≠ ∞ := by |
| apply ae_of_all |
| intro x |
| exact ENNReal.sum_ne_top.mpr (fun a _ => |
| ENNReal.mul_ne_top (measure_ne_top π _) ENNReal.ofReal_ne_top) |
| have hpost := finite_posterior_singletons π ν κ L hL hκ hpos hfinite |
| have hac : κ ∘ₘ π ≪ ν := by |
| rw [finite_observation_law_density π ν κ L hL hκ] |
| exact withDensity_absolutelyContinuous _ _ |
| filter_upwards [hpost, hac.ae_le hu₁, hac.ae_le hv₁] with x hx hux hvx |
| rw [integral_fintype Integrable.of_finite] |
| simp only [measureReal_def, hx, hπ, L, |
| binaryFusion_likelihood_sum hux.le hvx.le, smul_eq_mul, |
| ENNReal.toReal_div, ENNReal.toReal_mul, |
| ENNReal.toReal_ofReal (by norm_num : (0 : ℝ) ≤ 1 / 2), |
| ENNReal.toReal_ofReal (binaryFusion_denominator_pos hux hvx).le] |
| simp_rw [ENNReal.toReal_ofReal (binaryFusion_likelihood_nonneg hux.le hvx.le _)] |
| rw [sum_isingSpin] |
| simp only [spinSign_neg, spinSign_pos] |
| unfold binaryFusion |
| ring |
|
|
| theorem condExp_spin_eq_binaryFusion_of_likelihood |
| {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (π : Measure IsingSpin) [IsProbabilityMeasure π] |
| (hπ : ∀ a, π {a} = ENNReal.ofReal (1 / 2 : ℝ)) |
| (ν : Measure X) [SigmaFinite ν] |
| (κ : Kernel IsingSpin X) [IsFiniteKernel κ] |
| (S : Ω → IsingSpin) (Y : Ω → X) (hS : Measurable S) (hY : Measurable Y) |
| (hlaw : μ.map (fun ω => (S ω, Y ω)) = π ⊗ₘ κ) |
| (u v : X → ℝ) (hu : Measurable u) (hv : Measurable v) |
| (hu₁ : ∀ᵐ x ∂ν, |u x| < 1) (hv₁ : ∀ᵐ x ∂ν, |v x| < 1) |
| (hκ : ∀ a, κ a = ν.withDensity |
| (fun x => ENNReal.ofReal ((1 + spinSign a * u x) * (1 + spinSign a * v x)))) : |
| μ[fun ω => spinSign (S ω) | (inferInstance : MeasurableSpace X).comap Y] |
| =ᵐ[μ] fun ω => binaryFusion (u (Y ω)) (v (Y ω)) := by |
| exact FiniteBayes.condExp_state_eq_of_posterior_integral μ π κ S Y hS hY hlaw |
| spinSign (fun x => binaryFusion (u x) (v x)) |
| (posterior_spinMean_eq_binaryFusion π hπ ν κ u v hu hv hu₁ hv₁ hκ) |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
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|
