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ddc84c5 | 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 | import OAI.Analysis.FiniteFactor.Coupling.Marginals
import Mathlib.Probability.ProductMeasure
namespace OAI
namespace FiniteFactor
open MeasureTheory
open scoped ENNReal
noncomputable section
namespace CouplingParameters
/-- The augmented countable product retains each coordinate's label and pair. -/
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
/-- Every coordinatewise choice of endpoint marginals is exactly its product
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]
/-- Independence is recorded as the exact finite-cylinder product formula. -/
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
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