`Learning.IsAlgEnvSeq.isBayesAlgEnvSeq`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.isBayesAlgEnvSeq.{u_1, u_2, u_3, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} {Ω : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  [MeasurableSpace Ω] {Q : MeasureTheory.Measure 𝓔}
  [MeasureTheory.IsProbabilityMeasure Q]
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel κ] {alg : Algorithm 𝓐 𝓨}
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓔 × 𝓨} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (h :
    IsAlgEnvSeq A Y (Algorithm.prodLeft 𝓔 alg) (bayesStationaryEnv Q κ)
      P) :
  IsBayesAlgEnvSeq Q κ alg (fun ω => Prod.fst (Y 0 ω)) A
    (fun n ω => Prod.snd (Y n ω)) P
Learning.IsAlgEnvSeq.isBayesAlgEnvSeq.{u_1,
    u_2, u_3, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} {Ω : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] [MeasurableSpace Ω]
  {Q : MeasureTheory.Measure 𝓔}
  [MeasureTheory.IsProbabilityMeasure Q]
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel κ]
  {alg : Algorithm 𝓐 𝓨} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓔 × 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (h :
    IsAlgEnvSeq A Y
      (Algorithm.prodLeft 𝓔 alg)
      (bayesStationaryEnv Q κ) P) :
  IsBayesAlgEnvSeq Q κ alg
    (fun ω => Prod.fst (Y 0 ω)) A
    (fun n ω => Prod.snd (Y n ω)) P

Code

lemma IsAlgEnvSeq.isBayesAlgEnvSeq
    (h : IsAlgEnvSeq A Y (alg.prodLeft 𝓔) (bayesStationaryEnv Q κ) P) :
    IsBayesAlgEnvSeq Q κ alg (fun ω ↦ (Y 0 ω).1) A (fun n ω ↦ (Y n ω).2) P where
  measurable_param

Proof

(h.measurable_feedback 0).fst
  measurable_action := h.measurable_action
  measurable_feedback n := (h.measurable_feedback n).snd
  hasLaw_env := by
    apply HasCondDistrib.hasLaw_of_const
    simpa [bayesStationaryEnv] using h.hasCondDistrib_feedback_zero.fst
  hasCondDistrib_action_zero := by
    have hc : HasCondDistrib (fun ω ↦ (Y 0 ω).1) (A 0) (Kernel.const _ Q) P := by
      simpa [bayesStationaryEnv] using h.hasCondDistrib_feedback_zero.fst
    simpa [h.hasLaw_action_zero.map_eq, Algorithm.prodLeft] using hc.const_map_of_const
  hasCondDistrib_feedback_zero :=
    h.hasCondDistrib_feedback_zero.of_compProd.measurableEquiv_comp_right MeasurableEquiv.prodComm
  hasCondDistrib_action n := by
    let f : (Iic n → 𝓐 × 𝓔 × 𝓨) → 𝓔 × (Iic n → 𝓐 × 𝓨) :=
      fun h ↦ ((h ⟨0, by simp⟩).2.1, fun i ↦ ((h i).1, (h i).2.2))
    have hc : HasCondDistrib (A (n + 1)) (history A Y n)
        (((alg.policy n).comap Prod.snd (by fun_prop)).comap f (by fun_prop)) P :=
      h.hasCondDistrib_action n
    exact hc.comp_right (f := f)
  hasCondDistrib_feedback n := by
    let f : (Iic n → 𝓐 × 𝓔 × 𝓨) × 𝓐 → (Iic n → 𝓐 × 𝓨) × 𝓔 × 𝓐 :=
      fun p ↦ ((fun i ↦ ((p.1 i).1, (p.1 i).2.2)), (p.1 ⟨0, by simp⟩).2.1, p.2)
    have hc : HasCondDistrib (fun ω ↦ (Y (n + 1) ω).2)
        (fun ω ↦ (history A Y n ω, A (n + 1) ω))
        ((Kernel.prodMkLeft ((Iic n) → 𝓐 × 𝓨) κ).comap f (by fun_prop)) P := by
      simpa [bayesStationaryEnv, Kernel.prodMkLeft, ← Kernel.comap_comp_right, Function.comp_def]
        using (h.hasCondDistrib_feedback n).snd
    exact hc.comp_right

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 7 project declarations, 53 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IsAlgEnvSeq.isBayesAlgEnvSeq🔗

`Learning.IsAlgEnvSeq.isBayesAlgEnvSeq`🔗