| import OAI.Probability.InvariantIsing.Cavity.CavityCutoffWeight |
| import OAI.Probability.InvariantIsing.Cavity.CavityPhysicalGibbsSplit |
|
|
| namespace OAI |
|
|
| /-! Factoring a restricted cavity test over the base Gibbs law. The |
| cutoff remains inside the weight, so its zero convention is preserved. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set |
| open scoped Classical |
|
|
| namespace InvariantIsing |
|
|
| lemma cavity_reference_tilt_add {X : Type*} [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] |
| (μ : Measure X) [IsProbabilityMeasure μ] (J V : X → ℝ) |
| (hj : Integrable (fun x => Real.exp (J x)) μ) |
| (hfull : Integrable (fun x => Real.exp (J x + V x)) μ) |
| {r : ℕ} (F : (Fin r → X) → ℝ) : |
| referenceReplicaMean μ (fun x => J x + V x) F = |
| referenceReplicaMean (μ.tilted J) V F := by |
| let := isProbabilityMeasure_tilted hj |
| have hv : Integrable (fun x => Real.exp (V x)) (μ.tilted J) := by |
| rw [integrable_tilted_iff hj] |
| simpa only [smul_eq_mul, ← Real.exp_add] using hfull |
| rw [referenceReplicaMean_eq_tilted μ _ hfull, |
| referenceReplicaMean_eq_tilted (μ.tilted J) V hv, tilted_tilted hj] |
| rfl |
|
|
| lemma cavity_cutoff_tilt_add {X : Type*} [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] |
| (μ : Measure X) [IsProbabilityMeasure μ] (J V : X → ℝ) |
| (hj : Integrable (fun x => Real.exp (J x)) μ) |
| (hfull : Integrable (fun x => Real.exp (J x + V x)) μ) |
| (s : Set X) (F : (Fin 2 → X) → ℝ) : |
| cavityCutoffReplicaMean μ (fun x => J x + V x) s F = |
| cavityCutoffReplicaMean (μ.tilted J) V s F := by |
| unfold cavityCutoffReplicaMean |
| rw [cavity_reference_tilt_add μ J V hj hfull, |
| cavity_reference_tilt_add μ J V hj hfull] |
|
|
| theorem cavity_base_cutoff_indicator {X Y : Type*} |
| [MeasurableSpace X] [MeasurableSpace Y] |
| [Countable X] [Countable Y] [MeasurableSingletonClass X] [MeasurableSingletonClass Y] |
| (μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure Y) [IsProbabilityMeasure ν] |
| (J : X → ℝ) (V : X × Y → ℝ) |
| (hj : Integrable (fun x => Real.exp (J x)) μ) |
| (hfull : Integrable (fun x => Real.exp (J x.1 + V x)) (μ.prod ν)) |
| (s : Set (X × Y)) (F : (Fin 2 → X × Y) → ℝ) : |
| cavityCutoffReplicaMean (μ.prod ν) (fun x => J x.1 + V x) s F = |
| cavityWeightedReplicaMean ((μ.tilted J).prod ν) |
| (s.indicator (fun x => Real.exp (V x))) F := by |
| let := isProbabilityMeasure_tilted hj |
| have hv := cavity_base_product_tilt_integrable μ ν J (measurable_of_countable J) hj V hfull |
| rw [cavity_cutoff_tilt_add (μ.prod ν) (fun x => J x.1) V (hj.comp_fst ν) hfull, |
| ← cavity_tilt_prod_left μ ν J (measurable_of_countable J)] |
| exact cavity_cutoff_replica_indicator ((μ.tilted J).prod ν) V hv s F |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
