| import OAI.Probability.FreeIsing.Cavity.LogOdds |
| import OAI.Probability.FreeIsing.Estimates.ComponentFieldRecursion |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal BigOperators |
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| namespace Problem367 |
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| local instance : MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
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| theorem HasFreeIsingLaw.gaussian_component_field_sub_observation_le |
| {d : ℕ} (hd : 2 ≤ d) {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (t : ℝ≥0) (c : Fin d) : |
| let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s x => (t : ℝ) * spinSign s + x |
| ∀ᵐ z ∂ρ, |
| |Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c)) |
| (RootedBroadcast.branchRoot c) z) - |
| PlantedSymmetry.observation t z (RootedBroadcast.branchRoot c)| ≤ |
| ((d - 1 : ℕ) : ℝ) * beta := by |
| classical |
| filter_upwards [hμ.gaussian_component_field_recursion hd hb t c] with z hz |
| rw [hz, add_sub_cancel_left] |
| have hcard : Fintype.card {a : Fin d // c ≠ a} = d - 1 := by |
| simp [Fintype.card_subtype_compl] |
| calc |
| _ ≤ ∑ a : {a : Fin d // c ≠ a}, |
| |cavityMessage (Real.tanh beta) |
| (Real.artanh (selectedPosterior |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) |
| (fun s x => (t : ℝ) * spinSign s + x) (childComponentIndexVertex c a.1) |
| (prefixFlip c (RootedBroadcast.branchRoot a.1)) z))| := |
| Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _a : {a : Fin d // c ≠ a}, beta := |
| Finset.sum_le_sum (fun _ _ => abs_cavityMessage_tanh_le hb _) |
| _ = ((d - 1 : ℕ) : ℝ) * beta := by simp [hcard] |
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| theorem HasFreeIsingLaw.gaussian_component_field_recursion_cond_spin |
| {d : ℕ} (hd : 2 ≤ d) {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (t : ℝ≥0) (c : Fin d) (s : IsingSpin) : |
| let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun a x => (t : ℝ) * spinSign a + x |
| ∀ᵐ z ∂cond ρ {z | z.1 (RootedBroadcast.branchRoot c) = s}, |
| Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c)) |
| (RootedBroadcast.branchRoot c) z) = |
| (t : ℝ) * spinSign s + z.2 (RootedBroadcast.branchRoot c) + |
| ∑ a : {a : Fin d // c ≠ a}, cavityMessage (Real.tanh beta) |
| (Real.artanh (selectedPosterior ρ F (childComponentIndexVertex c a.1) |
| (prefixFlip c (RootedBroadcast.branchRoot a.1)) z)) := by |
| have hm : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| z.1 (RootedBroadcast.branchRoot c)) := |
| (measurable_pi_apply _).comp measurable_fst |
| have hset : MeasurableSet {z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) | |
| z.1 (RootedBroadcast.branchRoot c) = s} := hm (measurableSet_singleton s) |
| filter_upwards [cond_absolutelyContinuous.ae_le |
| (hμ.gaussian_component_field_recursion hd hb t c), ae_cond_mem hset] with z hz hs |
| rw [hz] |
| change z.1 (RootedBroadcast.branchRoot c) = s at hs |
| simp only [PlantedSymmetry.observation, hs] |
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| end Problem367 |
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| end |
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| end OAI |
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