LeanMachineLearning exposition

Learning.measurable_empMean'๐Ÿ”—

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Minimal Lean file

measurable_empMean'๐Ÿ”—

LemmaLearning.measurable_empMean'
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_empMean'.{u_1} {๐“ : Type u_1}
  {m๐“ : MeasurableSpace ๐“} [DecidableEq ๐“] [MeasurableSingletonClass ๐“]
  (n : โ„•) (a : ๐“) : Measurable fun h => empMean' n h a
Learning.measurable_empMean'.{u_1}
  {๐“ : Type u_1} {m๐“ : MeasurableSpace ๐“}
  [DecidableEq ๐“]
  [MeasurableSingletonClass ๐“] (n : โ„•)
  (a : ๐“) :
  Measurable fun h => empMean' n h a

Code

lemma measurable_empMean' [MeasurableSingletonClass ๐“] (n : โ„•) (a : ๐“) :
    Measurable (fun h โ†ฆ empMean' n h a)
Type uses (1)
Body uses (4)
Used by (2)

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Proof
by
  unfold empMean'
  fun_prop

Dependency graph

Type dependencies (1)

empMean'๐Ÿ”—

DefinitionLearning.empMean'
Details

Empirical mean of arm a at time n.

๐Ÿ”—def
Learning.empMean'.{u_1} {๐“ : Type u_1} [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) : โ„
Learning.empMean'.{u_1} {๐“ : Type u_1}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) :
  โ„

Code

noncomputable
def empMean' (n : โ„•) (h : Iic n โ†’ ๐“ ร— โ„) (a : ๐“) :=
  (sumRewards' n h a) / (pullCount' n h a)
Body uses (2)
Used by (18)

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All dependencies, transitively (2)

sumRewards'๐Ÿ”—

DefinitionLearning.sumRewards'
Details

Sum of rewards of arm a up to (and including) time n.

๐Ÿ”—def
Learning.sumRewards'.{u_1} {๐“ : Type u_1} [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) : โ„
Learning.sumRewards'.{u_1} {๐“ : Type u_1}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) :
  โ„

Code

noncomputable
def sumRewards' (n : โ„•) (h : Iic n โ†’ ๐“ ร— โ„) (a : ๐“) :=
  โˆ‘ s, if (h s).1 = a then (h s).2 else 0
Used by (9)

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pullCount'๐Ÿ”—

DefinitionLearning.pullCount'
Details

Number of pulls of arm a up to (and including) time n. This is the number of entries in h in which the arm is a.

๐Ÿ”—def
Learning.pullCount'.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•) (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) : โ„•
Learning.pullCount'.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) :
  โ„•

Code

noncomputable
def pullCount' (n : โ„•) (h : Iic n โ†’ ๐“ ร— R) (a : ๐“) := #{s | (h s).1 = a}
Used by (29)

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