| import OAI.Probability.PosteriorReplicas.Learners.LearnerModelBridge |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter Function |
| open scoped ENNReal NNReal RealInnerProductSpace unitInterval |
| noncomputable section |
| namespace PosteriorRegression |
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|
| lemma aemeasurable_completed_prod {X Y Z : Type*} [mX : MeasurableSpace X] [mY : MeasurableSpace Y] |
| [MeasurableSpace Z] [MeasurableSpace.CountablyGenerated Z] |
| (μ : Measure X) (ν : Measure Y) [SFinite ν] |
| {f : X × Y → Z} |
| (hf : @Measurable (NullMeasurableSpace X μ × Y) Z _ _ f) : |
| AEMeasurable f (μ.prod ν) := by |
| apply NullMeasurable.aemeasurable |
| have hp : @Measurable (NullMeasurableSpace (X × Y) (μ.prod ν)) |
| (NullMeasurableSpace X μ × Y) _ _ (fun z => (z.1,z.2)) := by |
| apply Measurable.prodMk |
| · intro s hs |
| have hs' : @NullMeasurableSet X mX s μ := hs |
| change @NullMeasurableSet (X × Y) _ (Prod.fst ⁻¹' s) (μ.prod ν) |
| exact @NullMeasurableSet.preimage (X × Y) X _ mX _ μ Prod.fst s hs' |
| (Measure.quasiMeasurePreserving_fst (μ := μ) (ν := ν)) |
| · exact measurable_snd.nullMeasurable.measurable' |
| exact hf.comp hp |
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| lemma aemeasurable_countable_choice {X J Y : Type*} [MeasurableSpace X] [MeasurableSpace J] |
| [MeasurableSpace Y] [Countable J] [MeasurableSingletonClass J] |
| {μ : Measure X} {j : X → J} (hj : AEMeasurable j μ) |
| {f : J → X → Y} (hf : ∀ a, AEMeasurable (f a) μ) : |
| AEMeasurable (fun x => f (j x) x) μ := by |
| let g : J → X → Y := fun a => (hf a).mk (f a) |
| have hg : Measurable (fun z : X × J => g z.2 z.1) := |
| measurable_from_prod_countable_left (fun a => (hf a).measurable_mk) |
| apply (hg.comp_aemeasurable (aemeasurable_id.prodMk hj)).congr |
| filter_upwards [ae_all_iff.mpr (fun a => (hf a).ae_eq_mk)] with x hx |
| exact (hx (j x)).symm |
|
|
| namespace Learner |
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|
| instance runStatus_measurableSingleton (M : ℕ) : MeasurableSingletonClass (RunStatus M) where |
| measurableSet_singleton r := by |
| rcases r with a | done |
| · simpa only [Set.image_singleton] using (measurableSet_inl_image (β := ℕ × State M) (s := {a})).mpr (measurableSet_singleton a) |
| · simpa only [Set.image_singleton] using (measurableSet_inr_image (α := State M) (s := {done})).mpr (measurableSet_singleton done) |
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| lemma uniform_experiment_of_steps {d M T : ℕ} {Ω : Type*} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hinit : Measurable L.initialChoice) |
| (htrans : ∀ t a (i : Fin T), AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => |
| L.transition t z.2.1 a (z.2.2 i,⟪z.2.2 i,z.1.val⟫)) |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T)))) |
| (hout : ∀ t a, Measurable (fun ω => L.output t ω a)) : |
| AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => L.estimate z.1 z.2.1 z.2.2) |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := by |
| let μ := (uniformSphere d).prod (ρ.prod (sampleLaw d T)) |
| have hstart : Measurable L.start := by |
| have he : L.start = (fun a : Bool × State M => if a.1 then Sum.inr (0,a.2) else Sum.inl a.2) ∘ L.initialChoice := rfl |
| rw [he] |
| exact (measurable_of_finite _).comp hinit |
| have hr (n : ℕ) : AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => L.run z.1 z.2.1 z.2.2 n) μ := by |
| induction n with |
| | zero => exact (hstart.comp measurable_snd.fst).aemeasurable |
| | succ n ih => |
| have hf (r : RunStatus M) : AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => |
| (match r with |
| | .inr done => .inr done |
| | .inl a => if hn : n<T then |
| let v := L.transition (n+1) z.2.1 a (z.2.2 ⟨n,hn⟩,⟪z.2.2 ⟨n,hn⟩,z.1.val⟫) |
| if v.1 then .inr (n+1,v.2) else .inl v.2 |
| else .inl a : RunStatus M)) μ := by |
| rcases r with a | done |
| · by_cases hn : n<T |
| · simp only [dite_eq_left hn] |
| exact (measurable_of_finite (fun v : Bool × State M => |
| if v.1 then Sum.inr (n+1,v.2) else Sum.inl v.2)).comp_aemeasurable (htrans (n+1) a ⟨n,hn⟩) |
| · simpa only [dite_eq_right hn] using (aemeasurable_const : AEMeasurable (fun _ : UnitSphere d × (Ω × Samples d T) => Sum.inl a : _ → RunStatus M) μ) |
| · exact aemeasurable_const |
| exact aemeasurable_countable_choice ih hf |
| have ht : AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => L.terminal z.1 z.2.1 z.2.2) μ := by |
| exact (measurable_of_countable (fun r : RunStatus M => |
| match r with | .inl a => (T,a) | .inr done => done)).comp_aemeasurable (hr T) |
| have ho (r : ℕ × State M) : AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => L.output r.1 z.2.1 r.2) μ := |
| ((hout r.1 r.2).comp measurable_snd.fst).aemeasurable |
| exact aemeasurable_countable_choice ht ho |
|
|
| end Learner |
| end PosteriorRegression |
| end |
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| end OAI |
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