File size: 3,534 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Cascade.PoissonLaplace
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy
variable {S : Type*} [MeasurableSpace S]
def indexedFiniteCloud (a : ℕ × (ℕ → S)) : Measure S :=
finitePointMeasure a.1 (fun i => a.2 i.val)
lemma indexedFiniteCloud_measurable : Measurable (indexedFiniteCloud (S := S)) := by
have h : Measurable (fun a : (ℕ → S) × ℕ => indexedFiniteCloud (a.2,a.1)) := by
apply measurable_from_prod_countable_left
intro n
exact (finitePointMeasure_measurable n).comp (Measurable.of_eval fun index => measurable_pi_apply index.val)
exact h.comp measurable_swap
lemma iid_prefix_law (ν : Measure S) [IsProbabilityMeasure ν] (n : ℕ) :
(Measure.infinitePi (fun _ : ℕ => ν)).map (fun x (i : Fin n) => x i.val) =
Measure.pi (fun _ : Fin n => ν) := by
rw [Measure.map_infinitePi_infinitePi_of_inj (f := Fin.val) Fin.val_injective,
Measure.infinitePi_eq_pi]
lemma indexedFiniteCloud_law (r : ℝ≥0) (ν : Measure S) [IsProbabilityMeasure ν] :
((poissonMeasure r).prod (Measure.infinitePi (fun _ : ℕ => ν))).map indexedFiniteCloud =
finitePoissonLaw r ν := by
rw [← Measure.sum_smul_dirac (poissonMeasure r),Measure.prod_sum_left,
Measure.map_sum indexedFiniteCloud_measurable.aemeasurable]
simp_rw [Measure.prod_smul_left,Measure.map_smul _ indexedFiniteCloud_measurable.aemeasurable,Measure.dirac_prod,
Measure.map_map indexedFiniteCloud_measurable measurable_prodMk_left]
unfold finitePoissonLaw
congr 1
funext n
congr 1
rw [← iid_prefix_law ν n,Measure.map_map (finitePointMeasure_measurable n) (by fun_prop)]
rfl
def indexedPoissonMeasure (a : ℕ → ℕ × (ℕ → S)) : Measure S :=
Measure.sum (fun i => indexedFiniteCloud (a i))
lemma indexedPoissonMeasure_measurable : Measurable (indexedPoissonMeasure (S := S)) :=
measureSum_measurable.comp (Measurable.of_eval fun index =>
indexedFiniteCloud_measurable.comp (measurable_pi_apply index))
lemma indexedPoissonMeasure_law (r : ℕ → ℝ≥0) (ν : ℕ → Measure S)
[∀ i, IsProbabilityMeasure (ν i)] :
(Measure.infinitePi (fun i => (poissonMeasure (r i)).prod
(Measure.infinitePi (fun _ : ℕ => ν i)))).map indexedPoissonMeasure =
countablePoissonLaw r ν := by
unfold indexedPoissonMeasure countablePoissonLaw
have hm : Measurable (fun a : ℕ → ℕ × (ℕ → S) => fun i => indexedFiniteCloud (a i)) :=
Measurable.of_eval fun index => indexedFiniteCloud_measurable.comp (measurable_pi_apply index)
change (Measure.infinitePi (fun i => (poissonMeasure (r i)).prod
(Measure.infinitePi (fun _ : ℕ => ν i)))).map
((fun η : ℕ → Measure S => Measure.sum η) ∘ (fun a : ℕ → ℕ × (ℕ → S) => fun i => indexedFiniteCloud (a i))) = _
rw [← Measure.map_map measureSum_measurable hm,Measure.infinitePi_map_pi]
· simp_rw [indexedFiniteCloud_law]
· exact fun _ => indexedFiniteCloud_measurable
lemma indexedPoissonMeasure_canonical_law [Nonempty S] (κ : Measure S) [SFinite κ] :
(Measure.infinitePi (fun i =>
(poissonMeasure (((sfiniteSeq κ i) univ).toNNReal)).prod
(Measure.infinitePi (fun _ : ℕ => finiteIntensityMarks (sfiniteSeq κ i))))).map
indexedPoissonMeasure = poissonRandomMeasureLaw κ :=
indexedPoissonMeasure_law _ _
end SphericalPerceptronFreeEnergy
end
end OAI
|