LeanMachineLearning exposition

Learning.IT.adapted_action๐Ÿ”—

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Minimal Lean file

adapted_action๐Ÿ”—

LemmaLearning.IT.adapted_action
Details

No docstring.

๐Ÿ”—theorem
Learning.IT.adapted_action.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ} :
  MeasureTheory.Adapted (IT.filtration ๐“ ๐“จ) action
Learning.IT.adapted_action.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} :
  MeasureTheory.Adapted
    (IT.filtration ๐“ ๐“จ) action

Code

lemma adapted_action : Adapted (IT.filtration ๐“ ๐“จ) action
Type uses (2)
Body uses (4)
Used by (3)

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Proof
by
  intro n
  rw [filtration_eq_comap, action_eq_eval_comp_hist]
  exact measurable_comp_comap _ (by fun_prop)

Dependency graph

Type dependencies (2)

filtration๐Ÿ”—

DefinitionLearning.IT.filtration
Details

Filtration of the algorithm Seq.

๐Ÿ”—def
Learning.IT.filtration.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] :
  MeasureTheory.Filtration โ„• inferInstance
Learning.IT.filtration.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] :
  MeasureTheory.Filtration โ„• inferInstance

Code

protected def filtration (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] :
    Filtration โ„• (inferInstance : MeasurableSpace (โ„• โ†’ ๐“ ร— ๐“จ)) :=
  MeasureTheory.Filtration.piLE (X := fun _ โ†ฆ ๐“ ร— ๐“จ)
Used by (13)

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action๐Ÿ”—

DefinitionLearning.IT.action
Details

action n is the action pulled at time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

๐Ÿ”—def
Learning.IT.action.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“
Learning.IT.action.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“

Code

def action (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“ := (h n).1
Used by (31)

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