LeanMachineLearning exposition

Learning.IT.adapted_step🔗

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Minimal Lean file

adapted_step🔗

LemmaLearning.IT.adapted_step
Details

No docstring.

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

Code

lemma adapted_step : Adapted (IT.filtration 𝓐 𝓨) (step (𝓐
Type uses (2)
Body uses (4)

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Proof
𝓐) (𝓨 := 𝓨)) := by
  intro n
  rw [filtration_eq_comap, step_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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step🔗

DefinitionLearning.IT.step
Details

Action and feedback at step n.

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

Code

def step (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓐 × 𝓨 := h n
Used by (13)

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