`Bandits.condDistrib_reward''`🔗

Lemma

No docstring.

theorem

Bandits.condDistrib_reward''.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {R : ℕ → Ω → ℝ} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] [Countable 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  (n : ℕ) :
  ⇑𝓛[fun ω => R n (Prod.fst ω) | fun ω => A n (Prod.fst ω);
        MeasureTheory.Measure.prod P
          (streamMeasure
            ν)] =ᵐ[MeasureTheory.Measure.map (fun ω => A n (Prod.fst ω))
      (MeasureTheory.Measure.prod P (streamMeasure ν))]
    ⇑ν
Bandits.condDistrib_reward''.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  [Countable 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  (n : ℕ) :
  ⇑𝓛[fun ω => R n (Prod.fst ω) | fun ω =>
        A n (Prod.fst ω);
        MeasureTheory.Measure.prod P
          (streamMeasure
            ν)] =ᵐ[MeasureTheory.Measure.map
      (fun ω => A n (Prod.fst ω))
      (MeasureTheory.Measure.prod P
        (streamMeasure ν))]
    ⇑ν

Code

lemma condDistrib_reward'' [Countable 𝓐]
    (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (n : ℕ) :
    𝓛[fun ω ↦ R n ω.1 | fun ω ↦ A n ω.1; 𝔓] =ᵐ[(𝔓).map (fun ω ↦ A n ω.1)] ν

Proof

by
  have hA := h.measurable_action
  have hR := h.measurable_feedback
  have h_ra' : 𝓛[R n | A n; P] =ᵐ[P.map (A n)] ν := h.condDistrib_feedback_stationaryEnv n
  have h_law : (𝔓).map (fun ω ↦ A n ω.1) = P.map (A n) := by
    change ((𝔓).map (A n ∘ Prod.fst)) = _
    rw [← Measure.map_map (by fun_prop) (by fun_prop), ← Measure.fst, Measure.fst_prod]
  rw [h_law]
  have h_prod : 𝓛[fun ω ↦ R n ω.1 | fun ω ↦ A n ω.1; 𝔓]
      =ᵐ[P.map (A n)] 𝓛[R n | A n; P] :=
    condDistrib_fst_prod _ (by fun_prop) _
  filter_upwards [h_ra', h_prod] with ω h_eq h_prod
  rw [h_prod, h_eq]

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: 8 project declarations, 80 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.

Bandits.condDistrib_reward''🔗

`Bandits.condDistrib_reward''`🔗