| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.Program |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler.Program |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {Ω E R : Type*} [MeasurableSpace Ω] [MeasurableSpace E] [MeasurableSpace R] {d q r : ℕ} |
|
|
| def independentSeq (A : Program Ω d q E) (B : Program (E × R) d r E) : |
| Program (Ω × R) d (q+r) E := |
| (A.seedMap Prod.fst measurable_fst).seq |
| (B.seedMap (fun p => (p.2,p.1.2)) (by fun_prop)) |
|
|
| lemma independentSeq_run (A : Program Ω d q E) (B : Program (E × R) d r E) |
| (V : Point d → ℝ) (p : Ω × R) : |
| (A.independentSeq B).run V p=B.run V (A.run V p.1,p.2) := by |
| rw [independentSeq,seq_run,seedMap_run,seedMap_run] |
|
|
| lemma independentSeq_law (A : Program Ω d q E) (B : Program (E × R) d r E) |
| {V : Point d → ℝ} (hV : Measurable (firstOrderReply V)) (μ : Measure Ω) (γ : Measure R) |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure γ] : |
| (μ.prod γ).map ((A.independentSeq B).run V)=((μ.map (A.run V)).prod γ).map (B.run V) := by |
| have hm := Measure.map_prod_map μ γ (A.measurable_run V hV) measurable_id |
| rw [Measure.map_id] at hm |
| rw [hm,Measure.map_map (B.measurable_run V hV) ((A.measurable_run V hV).prodMap measurable_id)] |
| congr 1 |
| funext p |
| exact independentSeq_run A B V p |
|
|
| end LogConcaveSampling.OracleCompiler.Program |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
