| import OAI.Probability.InvariantIsing.Fields.CascadeSeedCoupling |
| import OAI.Probability.IsingPerceptron.NoiseTreeKeep |
|
|
| namespace OAI |
|
|
| /-! Finite-depth vector cascade marking by independent uniform seeds. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| namespace InvariantIsing |
|
|
| def cascadeSeedLaw : ProbabilityMeasure unitInterval := ⟨volume, inferInstance⟩ |
|
|
| section Coordinates |
| variable {ι : Type} [Fintype ι] |
|
|
| def cascadeSeedTree : (n : ℕ) → (ℕ → ℝ) → |
| (ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) → (ι → ℝ) → |
| NoiseTree unitInterval n → NoiseTree (ι → ℝ) n |
| | 0, _, _, _, _ => PUnit.unit |
| | n+1, b, ψ, z, T => |
| (sigmaPart ((powerIntensity (b 0)).prod ((cascadeSeedLaw : Measure unitInterval).prod |
| (noiseCascadeLaw unitInterval n (fun j => b (j+1)) (fun _ => cascadeSeedLaw)))) T).map |
| (fun p => (p.1, (ψ 0 z p.2.1, |
| cascadeSeedTree n (fun j => b (j+1)) (fun j => ψ (j+1)) |
| (z+ψ 0 z p.2.1) p.2.2))) |
|
|
| omit [Fintype ι] in |
| lemma measurable_cascadeSeedTree (n : ℕ) (b : ℕ → ℝ) |
| (ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) |
| (hψ : ∀ i, Measurable (Function.uncurry (ψ i))) : |
| Measurable (fun p : (ι → ℝ) × NoiseTree unitInterval n => cascadeSeedTree n b ψ p.1 p.2) := by |
| induction n generalizing b ψ with |
| | zero => exact measurable_const |
| | succ n ih => |
| unfold cascadeSeedTree |
| apply measurable_map_sigmaPart_param _ measurable_snd |
| (G := fun p : ((ι → ℝ) × NoiseTree unitInterval (n+1)) × |
| (ℝ × (unitInterval × NoiseTree unitInterval n)) => |
| (p.2.1, (ψ 0 p.1.1 p.2.2.1, |
| cascadeSeedTree n (fun j => b (j+1)) (fun j => ψ (j+1)) |
| (p.1.1+ψ 0 p.1.1 p.2.2.1) p.2.2.2))) |
| have hmark : Measurable (fun p : ((ι → ℝ) × NoiseTree unitInterval (n+1)) × |
| (ℝ × (unitInterval × NoiseTree unitInterval n)) => ψ 0 p.1.1 p.2.2.1) := |
| (hψ 0).comp ((measurable_fst.comp measurable_fst).prodMk |
| (measurable_fst.comp (measurable_snd.comp measurable_snd))) |
| exact (by fun_prop : Measurable (fun p : ((ι → ℝ) × NoiseTree unitInterval (n+1)) × |
| (ℝ × (unitInterval × NoiseTree unitInterval n)) => p.2.1)).prodMk |
| (hmark.prodMk ((ih _ _ (fun i => hψ (i+1))).comp |
| (((by fun_prop : Measurable (fun p : ((ι → ℝ) × NoiseTree unitInterval (n+1)) × |
| (ℝ × (unitInterval × NoiseTree unitInterval n)) => p.1.1)).add hmark).prodMk (by fun_prop)))) |
|
|
| omit [Fintype ι] in |
| lemma cascadeSeedTree_law_succ (n : ℕ) (b : ℕ → ℝ) |
| (ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) |
| (hψ : ∀ i, Measurable (Function.uncurry (ψ i))) (z : ι → ℝ) : |
| let P : Measure (NoiseTree unitInterval n) := |
| noiseCascadeLaw unitInterval n (fun j => b (j+1)) (fun _ => cascadeSeedLaw) |
| let G := fun p : ℝ × (unitInterval × NoiseTree unitInterval n) => |
| (p.1, (ψ 0 z p.2.1, |
| cascadeSeedTree n (fun j => b (j+1)) (fun j => ψ (j+1)) (z+ψ 0 z p.2.1) p.2.2)) |
| (noiseCascadeLaw unitInterval (n+1) b (fun _ => cascadeSeedLaw) : |
| Measure (NoiseTree unitInterval (n+1))).map (cascadeSeedTree (n+1) b ψ z) = |
| poissonLaw (((powerIntensity (b 0)).prod ((cascadeSeedLaw : Measure unitInterval).prod P)).map G) := by |
| intro P G |
| let J := (powerIntensity (b 0)).prod ((cascadeSeedLaw : Measure unitInterval).prod P) |
| have hG : Measurable G := by |
| change Measurable (fun p : ℝ × (unitInterval × NoiseTree unitInterval n) => |
| (p.1, (ψ 0 z p.2.1, |
| cascadeSeedTree n (fun j => b (j+1)) (fun j => ψ (j+1)) (z+ψ 0 z p.2.1) p.2.2))) |
| have hmark : Measurable (fun p : ℝ × (unitInterval × NoiseTree unitInterval n) => ψ 0 z p.2.1) := |
| (hψ 0).comp (measurable_const.prodMk (measurable_fst.comp measurable_snd)) |
| exact measurable_fst.prodMk (hmark.prodMk |
| ((measurable_cascadeSeedTree n _ _ (fun i => hψ (i+1))).comp |
| ((measurable_const.add hmark).prodMk (by fun_prop)))) |
| rw [noiseCascadeLaw_succ] |
| change (poissonLaw J).map _ = poissonLaw (J.map G) |
| calc |
| _ = (poissonLaw J).map (fun T => T.map G) := by |
| apply Measure.map_congr |
| filter_upwards [sigmaPart_eq_ae J] with T hT |
| change (sigmaPart J T).map G = T.map G |
| rw [hT] |
| _ = _ := poissonLaw_map J hG |
|
|
| end Coordinates |
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
