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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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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