| 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 |
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|
