LeanMachineLearning exposition

Learning.integrable_pullCount๐Ÿ”—

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Minimal Lean file

integrable_pullCount๐Ÿ”—

LemmaLearning.integrable_pullCount
Details

No docstring.

๐Ÿ”—theorem
Learning.integrable_pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  {m๐“ : MeasurableSpace ๐“} {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {P : MeasureTheory.Measure ฮฉ} [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} [MeasurableSingletonClass ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n)) (a : ๐“) (n : โ„•) :
  MeasureTheory.Integrable (fun ฯ‰ => โ†‘(pullCount A a n ฯ‰)) P
Learning.integrable_pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  {m๐“ : MeasurableSpace ๐“}
  {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  [MeasurableSingletonClass ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (a : ๐“) (n : โ„•) :
  MeasureTheory.Integrable
    (fun ฯ‰ => โ†‘(pullCount A a n ฯ‰)) P

Code

lemma integrable_pullCount [MeasurableSingletonClass ๐“]
    (hA : โˆ€ n, Measurable (A n)) (a : ๐“) (n : โ„•) :
    Integrable (fun ฯ‰ โ†ฆ (pullCount A a n ฯ‰ : โ„)) P
Type uses (1)
Body uses (2)
Used by (2)

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Proof
by
  refine integrable_of_le_of_le (gโ‚ := 0) (gโ‚‚ := fun _ โ†ฆ n) (by fun_prop)
    (ae_of_all _ fun ฯ‰ โ†ฆ by simp) (ae_of_all _ fun ฯ‰ โ†ฆ ?_) (integrable_const _) (integrable_const _)
  simp only [Nat.cast_le]
  exact pullCount_le a n ฯ‰

Dependency graph

Type dependencies (1)

pullCount๐Ÿ”—

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

๐Ÿ”—def
Learning.pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„•
Learning.pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„•

Code

noncomputable
def pullCount (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„• :=
  #(filter (fun s โ†ฆ A s ฯ‰ = a) (range t))
Used by (146)

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