LeanMachineLearning exposition

Learning.IsBayesAlgEnvSeq๐Ÿ”—

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IsBayesAlgEnvSeq๐Ÿ”—

StructureLearning.IsBayesAlgEnvSeq
Details

IsBayesAlgEnvSeq Q ฮบ alg E A Y P states that there is a measure P : Measure ฮฉ such that the parameter E : ฮฉ โ†’ ๐“” has law Q and that the sequences of actions A : โ„• โ†’ ฮฉ โ†’ ๐“ and feedbacks Y : โ„• โ†’ ฮฉ โ†’ ๐“จ are generated by the algorithm alg : Algorithm ๐“ ๐“จ interacting with an underlying environment that depends on E and ฮบ (stationaryEnv (ฮบ.sectR (E ฯ‰))).

๐Ÿ”—structure
Learning.IsBayesAlgEnvSeq.{u_1, u_2, u_3, u_4} {๐“” : Type u_1}
  {๐“ : Type u_2} {๐“จ : Type u_3} {ฮฉ : Type u_4} [MeasurableSpace ๐“”]
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] [MeasurableSpace ฮฉ]
  (Q : MeasureTheory.Measure ๐“”) (ฮบ : ProbabilityTheory.Kernel (๐“” ร— ๐“) ๐“จ)
  (alg : Algorithm ๐“ ๐“จ) (E : ฮฉ โ†’ ๐“”) (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsBayesAlgEnvSeq.{u_1, u_2, u_3,
    u_4}
  {๐“” : Type u_1} {๐“ : Type u_2}
  {๐“จ : Type u_3} {ฮฉ : Type u_4}
  [MeasurableSpace ๐“”] [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] [MeasurableSpace ฮฉ]
  (Q : MeasureTheory.Measure ๐“”)
  (ฮบ : ProbabilityTheory.Kernel (๐“” ร— ๐“) ๐“จ)
  (alg : Algorithm ๐“ ๐“จ) (E : ฮฉ โ†’ ๐“”)
  (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsBayesAlgEnvSeq
    (Q : Measure ๐“”) (ฮบ : Kernel (๐“” ร— ๐“) ๐“จ) (alg : Algorithm ๐“ ๐“จ)
    (E : ฮฉ โ†’ ๐“”) (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ)
    (P : Measure ฮฉ) [IsFiniteMeasure P] : Prop where
  measurable_param : Measurable E := by fun_prop
  measurable_action n : Measurable (A n) := by fun_prop
  measurable_feedback n : Measurable (Y n) := by fun_prop
  hasLaw_env : HasLaw E Q P
  hasCondDistrib_action_zero : HasCondDistrib (A 0) E (Kernel.const _ alg.p0) P
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (fun ฯ‰ โ†ฆ (E ฯ‰, A 0 ฯ‰)) ฮบ P
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (fun ฯ‰ โ†ฆ (E ฯ‰, history A Y n ฯ‰))
      ((alg.policy n).prodMkLeft _) P
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ฯ‰ โ†ฆ (history A Y n ฯ‰, E ฯ‰, A (n + 1) ฯ‰))
      (ฮบ.prodMkLeft _) P
Type uses (2)
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Dependency graph

Type dependencies (2)

Algorithm๐Ÿ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

๐Ÿ”—structure
Learning.Algorithm.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Algorithm (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : โ„•) โ†’ Kernel (Iic n โ†’ ๐“ ร— ๐“จ) ๐“
  /-- The policy is a Markov kernel. -/
  [h_policy : โˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure ๐“
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
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history๐Ÿ”—

DefinitionLearning.history
Details

History of the algorithm-environment sequence up to time n.

๐Ÿ”—def
Learning.history.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ
Learning.history.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ

Code

def history (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) : Iic n โ†’ ๐“ ร— ๐“จ :=
  fun i โ†ฆ (A i ฯ‰, Y i ฯ‰)
Used by (72)

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