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a2ee19a | 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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.ProjectionMoments.BorelRules
import OAI.Probability.MemoryPrecision.AmbientGram
import OAI.Probability.MemoryPrecision.KernelModel
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
lemma aemeasurable_countable_select {X J Y : Type*} [MeasurableSpace X]
[MeasurableSpace J] [MeasurableSingletonClass J] [Countable J] [MeasurableSpace Y]
{μ : Measure X} {h : J → X → Y} (hh : ∀ j, AEMeasurable (h j) μ)
{φ : X → J} (hφ : AEMeasurable φ μ) : AEMeasurable (fun x => h (φ x) x) μ := by
have hmk : Measurable (fun x => (hh (hφ.mk φ x)).mk (h (hφ.mk φ x)) x) :=
(measurable_from_prod_countable_right
(f := fun z : J × X => (hh z.1).mk (h z.1) z.2)
(fun j => (hh j).measurable_mk)).comp (hφ.measurable_mk.prodMk measurable_id)
apply hmk.aemeasurable.congr
have hall : ∀ᵐ x ∂μ, ∀ j, h j x = (hh j).mk (h j) x :=
ae_all_iff.mpr (fun j => (hh j).ae_eq_mk)
filter_upwards [hφ.ae_eq_mk, hall] with x hx hx'
rw [← hx, ← hx']
lemma prior_observation_with_seed {X : Type v} [MeasurableSpace X]
{d T : ℕ} (ν : Measure X) [SFinite ν] {Ω : Type u} [MeasurableSpace Ω]
(ρ : Measure Ω) [SFinite ρ] (ι : X → UnitSphere d) (i : Fin T)
(ho : Measure.QuasiMeasurePreserving
(fun z : X × Samples d T => (z.2 i,⟪(ι z.1).val,z.2 i⟫))
(ν.prod (sampleLaw d T)) (observationReference d)) :
Measure.QuasiMeasurePreserving
(fun z : X × (Ω × Samples d T) => (z.2.1,(z.2.2 i,⟪(ι z.1).val,z.2.2 i⟫)))
(ν.prod (ρ.prod (sampleLaw d T))) (ρ.prod (observationReference d)) := by
let : IsProbabilityMeasure (sampleLaw d T) := by unfold sampleLaw; infer_instance
let : SigmaFinite (observationReference d) := by unfold observationReference; infer_instance
have h1 := (MeasurePreserving.id ν).prod
(Measure.measurePreserving_swap (μ := ρ) (ν := sampleLaw d T))
have h2 := (measurePreserving_prodAssoc ν (sampleLaw d T) ρ).symm
(MeasurableEquiv.prodAssoc : (X × Samples d T) × Ω ≃ᵐ X × (Samples d T × Ω))
have h3 := (Measure.measurePreserving_swap (μ := ν.prod (sampleLaw d T)) (ν := ρ)).comp (h2.comp h1)
have h4 := MeasureTheory.QuasiMeasurePreserving.prodMap (MeasurePreserving.id ρ).quasiMeasurePreserving ho
convert h4.comp h3.quasiMeasurePreserving using 1
rfl
lemma prior_estimate_aemeasurable {X : Type v} [MeasurableSpace X]
{d M T : ℕ} (ν : Measure X) [SFinite ν] {Ω : Type u} [MeasurableSpace Ω]
(ρ : Measure Ω) [SFinite ρ] (ι : X → UnitSphere d)
(L : Learner d M T Ω) (hL : L.Admissible ρ)
(ho : ∀ i : Fin T, Measure.QuasiMeasurePreserving
(fun z : X × Samples d T => (z.2 i,⟪(ι z.1).val,z.2 i⟫))
(ν.prod (sampleLaw d T)) (observationReference d)) :
AEMeasurable (fun z : X × (Ω × Samples d T) => L.estimate (ι z.1) z.2.1 z.2.2)
(ν.prod (ρ.prod (sampleLaw d T))) := by
let μ := ν.prod (ρ.prod (sampleLaw d T))
let : IsProbabilityMeasure (sampleLaw d T) := by unfold sampleLaw; infer_instance
have hω : Measure.QuasiMeasurePreserving (fun z : X × (Ω × Samples d T) => z.2.1) μ ρ :=
Measure.quasiMeasurePreserving_fst.comp Measure.quasiMeasurePreserving_snd
have ht (t : ℕ) (a : State M) (i : Fin T) :
AEMeasurable (fun z : X × (Ω × Samples d T) =>
L.transition t z.2.1 a (z.2.2 i,⟪z.2.2 i,(ι z.1).val⟫)) μ := by
have h := (hL.transition t a).comp_quasiMeasurePreserving
(prior_observation_with_seed ν ρ ι i (ho i))
simpa only [Function.comp_def, μ, real_inner_comm] using h
have hr (n : ℕ) : AEMeasurable
(fun z : X × (Ω × Samples d T) => L.run (ι z.1) z.2.1 z.2.2 n) μ := by
induction n with
| zero =>
exact (measurable_of_countable (fun a : Bool × State M =>
if a.1 then Sum.inr (0,a.2) else Sum.inl a.2)).comp_aemeasurable
(hL.initialChoice.comp_quasiMeasurePreserving hω)
| succ n ih =>
let g : Learner.RunStatus M → (X × (Ω × Samples d T)) → Learner.RunStatus M := fun j z =>
match j with
| .inr done => .inr done
| .inl a => if hn : n < T then
let b := L.transition (n+1) z.2.1 a (z.2.2 ⟨n,hn⟩,⟪z.2.2 ⟨n,hn⟩,(ι z.1).val⟫)
if b.1 then .inr (n+1,b.2) else .inl b.2
else .inl a
apply aemeasurable_countable_select (h := g) ?_ ih
intro j
cases j with
| inr done => exact aemeasurable_const
| inl a =>
by_cases hn : n < T
· simpa only [g, dite_eq_left hn, Function.comp_def] using
(measurable_of_countable (fun b : Bool × State M =>
if b.1 then Sum.inr (n+1,b.2) else Sum.inl b.2)).comp_aemeasurable (ht (n+1) a ⟨n,hn⟩)
· simpa only [g, dite_eq_right hn] using (aemeasurable_const (b := Sum.inl a))
have hterm : AEMeasurable
(fun z : X × (Ω × Samples d T) => L.terminal (ι z.1) z.2.1 z.2.2) μ := by
convert
(measurable_of_countable (fun j : Learner.RunStatus M =>
(match j with | .inl a => (T,a) | .inr p => p : ℕ × State M))).comp_aemeasurable (hr T) using 1
rfl
exact aemeasurable_countable_select
(fun j : ℕ × State M => (hL.output j.1 j.2).comp_quasiMeasurePreserving hω) hterm
end MemoryPrecision
end
end OAI
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