`Bandits.ArrayModel.condIndepFun_reward_hist`🔗

Lemma

The reward at time n + 1 is conditionally independent of the history up to time n, given the action at time n + 1 and the number of times that action has been pulled before time n + 1.

theorem

Bandits.ArrayModel.condIndepFun_reward_hist.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  [Nonempty 𝓐] [StandardBorelSpace 𝓐] [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace R] [Nonempty R] (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] (n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap
      (fun ω =>
        (action alg (n + 1) ω,
          Learning.pullCount (action alg) (action alg (n + 1) ω) (n + 1)
            ω))
      inferInstance)
    ⋯ (reward alg (n + 1)) (fun x => hist alg x n) (arrayMeasure ν)
Bandits.ArrayModel.condIndepFun_reward_hist.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  [Countable 𝓐] [StandardBorelSpace R]
  [Nonempty R]
  (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν]
  (n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap
      (fun ω =>
        (action alg (n + 1) ω,
          Learning.pullCount (action alg)
            (action alg (n + 1) ω) (n + 1)
            ω))
      inferInstance)
    ⋯ (reward alg (n + 1))
    (fun x => hist alg x n)
    (arrayMeasure ν)

Code

lemma condIndepFun_reward_hist (alg : Algorithm 𝓐 R) (ν : Kernel 𝓐 R) [IsMarkovKernel ν] (n : ℕ) :
    (reward alg (n + 1)) ⟂ᵢ[(fun ω ↦ (action alg (n + 1) ω,
          pullCount (action alg) (action alg (n + 1) ω) (n + 1) ω)),
        Measurable.prodMk (by fun_prop) (measurable_pullCount_action_add_one alg n);
        arrayMeasure ν]
      (hist alg · n)

Proof

by
  have h_cond := hasCondDistrib_reward_hist_action_pullCount alg ν n
  refine condIndepFun_of_exists_condDistrib_prod_ae_eq_prodMkLeft (by fun_prop) (by fun_prop) ?_
    h_cond.condDistrib_eq
  exact Measurable.prodMk (by fun_prop) (measurable_pullCount_action_add_one alg n)

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 5th 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: 19 project declarations, 140 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.ArrayModel.condIndepFun_reward_hist🔗

`Bandits.ArrayModel.condIndepFun_reward_hist`🔗