LeanMachineLearning exposition

Learning.IT.filtration_eq_comap๐Ÿ”—

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Minimal Lean file

filtration_eq_comap๐Ÿ”—

LemmaLearning.IT.filtration_eq_comap
Details

No docstring.

๐Ÿ”—theorem
Learning.IT.filtration_eq_comap.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ} (n : โ„•) :
  โ†‘(IT.filtration ๐“ ๐“จ) n = MeasurableSpace.comap (hist n) inferInstance
Learning.IT.filtration_eq_comap.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (n : โ„•) :
  โ†‘(IT.filtration ๐“ ๐“จ) n =
    MeasurableSpace.comap (hist n)
      inferInstance

Code

lemma filtration_eq_comap (n : โ„•) :
    IT.filtration ๐“ ๐“จ n = MeasurableSpace.comap (hist n) inferInstance
Type uses (2)
Body uses (1)
Used by (5)

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Proof
by
  simp [IT.filtration, Filtration.piLE_eq_comap_frestrictLe, โ† hist_eq_frestrictLe]

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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hist๐Ÿ”—

DefinitionLearning.IT.hist
Details

hist n is the history up to time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

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

Code

def hist (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : Iic n โ†’ ๐“ ร— ๐“จ := fun i โ†ฆ h i
Used by (23)

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