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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 | import OAI.Probability.InvariantIsing.Fields.CascadeSeedTree
namespace OAI
/-! The full retained finite vector cascade is realized by ancestor-dependent seeds. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal ENNReal
namespace InvariantIsing
theorem cascadeSeedTree_retained_law {ι : Type} [Fintype ι]
(n : ℕ) (b : ℕ → ℝ) (μ : ℕ → ProbabilityMeasure (ι → ℝ))
(c : ℕ → (ι → ℝ) × (ι → ℝ) → ℝ)
(ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ))
(hc : ∀ i, Measurable (c i)) (hψ : ∀ i, Measurable (Function.uncurry (ψ i)))
(hcpos : ∀ i z a, 0 < c i (z,a))
(hm : ∀ i z, (∫⁻ a, ENNReal.ofReal (c i (z,a)^b i) ∂(μ i : Measure (ι → ℝ))) = 1)
(hseed : ∀ i z, volume.map (ψ i z) = (μ i : Measure (ι → ℝ)).withDensity
(fun a => ENNReal.ofReal (c i (z,a)^b i))) (z : ι → ℝ) :
(noiseCascadeLaw unitInterval n b (fun _ => cascadeSeedLaw) :
Measure (NoiseTree unitInterval n)).map (cascadeSeedTree n b ψ z) =
(noiseCascadeLaw (ι → ℝ) n b μ : Measure (NoiseTree (ι → ℝ) n)).map
(noiseTreeKeep n b μ c (fun _ p => p.1+p.2) z) := by
induction n generalizing b μ c ψ z with
| zero =>
change (Measure.dirac (PUnit.unit : PUnit)).map (fun _ => PUnit.unit) =
(Measure.dirac (PUnit.unit : PUnit)).map (fun _ => PUnit.unit)
rfl
| succ n ih =>
let bs := fun j => b (j+1)
let ms := fun j => μ (j+1)
let cs := fun j => c (j+1)
let ps := fun j => ψ (j+1)
let Q : Measure (NoiseTree unitInterval n) :=
noiseCascadeLaw unitInterval n bs (fun _ => cascadeSeedLaw)
let P : Measure (NoiseTree (ι → ℝ) n) := noiseCascadeLaw (ι → ℝ) n bs ms
let ρ : Measure (ι → ℝ) := (μ 0 : Measure (ι → ℝ)).withDensity
(fun a => ENNReal.ofReal (c 0 (z,a)^b 0))
let : IsProbabilityMeasure ρ := ⟨by
change ((μ 0 : Measure (ι → ℝ)).withDensity _ ) Set.univ = 1
rw [withDensity_apply _ MeasurableSet.univ, setLIntegral_univ]
exact hm 0 z⟩
let : IsProbabilityMeasure Q := by
change IsProbabilityMeasure (noiseCascadeLaw unitInterval n bs (fun _ => cascadeSeedLaw) : Measure (NoiseTree unitInterval n))
infer_instance
let : IsProbabilityMeasure P := by
change IsProbabilityMeasure (noiseCascadeLaw (ι → ℝ) n bs ms : Measure (NoiseTree (ι → ℝ) n))
infer_instance
let G : (ι → ℝ) × NoiseTree unitInterval n → NoiseTree (ι → ℝ) n :=
fun p => cascadeSeedTree n bs ps (z+p.1) p.2
let H : (ι → ℝ) × NoiseTree (ι → ℝ) n → NoiseTree (ι → ℝ) n :=
fun p => noiseTreeKeep n bs ms cs (fun _ p => p.1+p.2) (z+p.1) p.2
have hG : Measurable G := (measurable_cascadeSeedTree n bs ps (fun i => hψ (i+1))).comp
((measurable_const.add measurable_fst).prodMk measurable_snd)
have hH : Measurable H :=
(measurable_noiseTreeKeep n bs ms (fun i => hc (i+1))
(fun _ => measurable_fst.add measurable_snd)).comp
((measurable_const.add measurable_fst).prodMk measurable_snd)
have hsub a : Q.map (fun q => G (a,q)) = P.map (fun p => H (a,p)) :=
ih bs ms cs ps (fun i => hc (i+1)) (fun i => hψ (i+1))
(fun i => hcpos (i+1)) (fun i => hm (i+1)) (fun i => hseed (i+1)) (z+a)
have hf : Measurable (ψ 0 z) := (hψ 0).comp (measurable_const.prodMk measurable_id)
have hlaw : (cascadeSeedLaw : Measure unitInterval).map (ψ 0 z) = ρ := hseed 0 z
have hpair := ancestor_fiber_map_product (cascadeSeedLaw : Measure unitInterval) ρ Q P hf hlaw hG hH hsub
have hmap : Measurable (fun p : unitInterval × NoiseTree unitInterval n =>
(ψ 0 z p.1,G (ψ 0 z p.1,p.2))) :=
(hf.comp measurable_fst).prodMk (hG.comp ((hf.comp measurable_fst).prodMk measurable_snd))
rw [cascadeSeedTree_law_succ n b ψ hψ z,
noiseTreeKeep_law_succ n b μ (u := fun (_ : ℕ) (p : (ι → ℝ) × (ι → ℝ)) => p.1+p.2)
hc (fun _ => measurable_fst.add measurable_snd) hcpos z]
apply poissonLaw_congr
change (((powerIntensity (b 0)).prod ((cascadeSeedLaw : Measure unitInterval).prod Q)).map
(Prod.map id (fun p => (ψ 0 z p.1,G (ψ 0 z p.1,p.2))))) =
(powerIntensity (b 0)).prod
((((μ 0 : Measure (ι → ℝ)).prod P).withDensity
(fun p => ENNReal.ofReal (c 0 (z,p.1)^b 0))).map (fun p => (p.1,H p)))
rw [← Measure.map_prod_map _ _ measurable_id hmap, Measure.map_id, hpair]
congr 1
congr 1
exact prod_withDensity_left
((((hc 0).comp (measurable_const.prodMk measurable_id)).pow_const (b 0)).ennreal_ofReal)
end InvariantIsing
end
end OAI
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