| import OAI.Analysis.FiniteFactor.Coupling.Marginals |
| import Mathlib.Probability.ProductMeasure |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| namespace CouplingParameters |
|
|
| / |
| abbrev CouplingSample := ℕ → Fin 6 × (ℝ × ℝ) |
|
|
| def productCoupling (p : ℕ → CouplingParameters) : Measure CouplingSample := |
| Measure.infinitePi (fun i => (p i).labelledCoupling) |
| |
| instance productCoupling_probability (p : ℕ → CouplingParameters) : |
| IsProbabilityMeasure (productCoupling p) := by |
| unfold productCoupling |
| infer_instance |
| |
| def productEndpoint (right : ℕ → Bool) (ω : CouplingSample) (i : ℕ) : ℝ := |
| endpoint (right i) (ω i).2 |
| |
| theorem productEndpoint_measurable (right : ℕ → Bool) : Measurable (productEndpoint right) := by |
| apply Measurable.of_eval |
| intro i |
| exact (endpoint_measurable (right i)).comp (measurable_snd.comp (measurable_pi_apply i)) |
| |
| theorem labelled_endpoint_density (p : CouplingParameters) (right : Bool) : |
| p.labelledCoupling.map (fun z => endpoint right z.2) = |
| unitIntervalMeasure.withDensity (fun t => ENNReal.ofReal (p.endpointDensity right t)) := by |
| rw [← p.coupling_endpoint_density] |
| unfold coupling |
| rw [Measure.map_map (endpoint_measurable right) measurable_snd] |
| rfl |
| |
| / |
| law, including the mixed prefixes used in biased-identity. -/ |
| theorem productEndpoint_law (p : ℕ → CouplingParameters) (right : ℕ → Bool) : |
| (productCoupling p).map (productEndpoint right) = |
| Measure.infinitePi (fun i => unitIntervalMeasure.withDensity |
| (fun t => ENNReal.ofReal ((p i).endpointDensity (right i) t))) := by |
| unfold productCoupling productEndpoint |
| rw [Measure.infinitePi_map_pi (fun i => (p i).labelledCoupling) |
| (f := fun i (z : Fin 6 × (ℝ × ℝ)) => endpoint (right i) z.2) (fun i => (endpoint_measurable (right i)).comp measurable_snd)] |
| simp_rw [labelled_endpoint_density] |
| |
| / |
| theorem productCoupling_cylinder (p : ℕ → CouplingParameters) (I : Finset ℕ) |
| (S : ℕ → Set (Fin 6 × (ℝ × ℝ))) (hS : ∀ i ∈ I, MeasurableSet (S i)) : |
| productCoupling p (Set.pi (I : Set ℕ) S) = ∏ i ∈ I, (p i).labelledCoupling (S i) := |
| Measure.infinitePi_pi _ hS |
| |
| end CouplingParameters |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
