LeanMachineLearning exposition

Learning.IT.adapted_feedback🔗

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Minimal Lean file

adapted_feedback🔗

LemmaLearning.IT.adapted_feedback
Details

No docstring.

🔗theorem
Learning.IT.adapted_feedback.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} :
  MeasureTheory.Adapted (IT.filtration 𝓐 𝓨) feedback
Learning.IT.adapted_feedback.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} :
  MeasureTheory.Adapted
    (IT.filtration 𝓐 𝓨) feedback

Code

lemma adapted_feedback : Adapted (IT.filtration 𝓐 𝓨) feedback
Type uses (2)
Body uses (4)

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

Dependency graph

Type dependencies (2)

filtration🔗

DefinitionLearning.IT.filtration
Details

Filtration of the algorithm Seq.

🔗def
Learning.IT.filtration.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] :
  MeasureTheory.Filtration ℕ inferInstance
Learning.IT.filtration.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] :
  MeasureTheory.Filtration ℕ inferInstance

Code

protected def filtration (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] :
    Filtration ℕ (inferInstance : MeasurableSpace (ℕ → 𝓐 × 𝓨)) :=
  MeasureTheory.Filtration.piLE (X := fun _ ↦ 𝓐 × 𝓨)
Used by (13)

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feedback🔗

DefinitionLearning.IT.feedback
Details

feedback n is the feedback at time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

🔗def
Learning.IT.feedback.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓨
Learning.IT.feedback.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓨

Code

def feedback (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓨 := (h n).2
Used by (16)

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