File size: 3,550 Bytes
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 | import OAI.Probability.InvariantIsing.Fields.CascadeSeedTree
import OAI.Probability.IsingPerceptron.NoiseGibbsRegular
namespace OAI
/-! Regularity of uniform-seed trees, independent of the ancestor state. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped ENNReal
namespace InvariantIsing
def CascadeSeedRegular (n : ℕ) (b : ℕ → ℝ) (T : NoiseTree unitInterval n) : Prop :=
NoiseGibbsRegular n b (fun _ => cascadeSeedLaw)
(fun _ (_ : PUnit × unitInterval) => 1) (fun _ _ => PUnit.unit) PUnit.unit T
lemma cascadeSeedRegular_ae (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) :
∀ᵐ T ∂(noiseCascadeLaw unitInterval n b (fun _ => cascadeSeedLaw) : Measure _),
CascadeSeedRegular n b T := by
exact noiseGibbsRegular_ae n b hb (fun _ => cascadeSeedLaw)
(fun _ => measurable_const) (fun _ => measurable_const) (fun _ _ _ => zero_lt_one)
(fun _ _ => by simp) PUnit.unit
lemma cascadeSeedRegular_sigma {n : ℕ} {b : ℕ → ℝ} {T : NoiseTree unitInterval (n+1)}
(hT : CascadeSeedRegular (n+1) b T) :
sigmaPart ((powerIntensity (b 0)).prod ((cascadeSeedLaw : Measure unitInterval).prod
(noiseCascadeLaw unitInterval n (fun j => b (j+1)) (fun _ => cascadeSeedLaw)))) T = T :=
Marked.sigmaPart_eq_of_good _ hT.2.2.1
lemma cascadeSeedRegular_tail {n : ℕ} {b : ℕ → ℝ} {T : NoiseTree unitInterval (n+1)}
(hT : CascadeSeedRegular (n+1) b T) :
∀ᵐ p ∂T, CascadeSeedRegular n (fun j => b (j+1)) p.2.2 := by
have h := hT.2.2.2
rwa [cascadeSeedRegular_sigma hT] at h
lemma cascadeSeedRegular_total {n : ℕ} {b : ℕ → ℝ} {T : NoiseTree unitInterval (n+1)}
(hT : CascadeSeedRegular (n+1) b T) :
0 < noiseTreeTotal unitInterval (n+1) T ∧ noiseTreeTotal unitInterval (n+1) T < ∞ := hT.1
theorem cascadeSeedTree_total {ι : Type} [Fintype ι] (n : ℕ) (b : ℕ → ℝ)
(ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ))
(hψ : ∀ i, Measurable (Function.uncurry (ψ i)))
(T : NoiseTree unitInterval n) (hT : CascadeSeedRegular n b T) (z : ι → ℝ) :
noiseTreeTotal (ι → ℝ) n (cascadeSeedTree n b ψ z T) = noiseTreeTotal unitInterval n T := by
induction n generalizing b ψ z with
| zero => rfl
| succ n ih =>
have hmap : 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))) := by
have hm : 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 (hm.prodMk
((measurable_cascadeSeedTree n _ _ (fun i => hψ (i+1))).comp
((measurable_const.add hm).prodMk (by fun_prop))))
rw [cascadeSeedTree, cascadeSeedRegular_sigma hT, noiseTreeTotal_succ, noiseTreeTotal_succ]
rw [lintegral_map (show Measurable (fun p : ℝ × ((ι → ℝ) × NoiseTree (ι → ℝ) n) =>
ENNReal.ofReal (max p.1 0) * noiseTreeTotal (ι → ℝ) n p.2.2) from
(by fun_prop : Measurable (fun p : ℝ × ((ι → ℝ) × NoiseTree (ι → ℝ) n) =>
ENNReal.ofReal (max p.1 0))).mul ((measurable_noiseTreeTotal (ι → ℝ) n).comp
(measurable_snd.comp measurable_snd))) hmap]
apply lintegral_congr_ae
filter_upwards [cascadeSeedRegular_tail hT] with p hp
rw [ih _ _ (fun i => hψ (i+1)) _ hp]
end InvariantIsing
end
end OAI
|