LeanMachineLearning exposition

Learning.feedbackCondActionπŸ”—

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feedbackCondActionπŸ”—

DefinitionLearning.feedbackCondAction
Details

The kernel representing the conditional distribution of the feedback given the action at time n in an oblivious environment.

πŸ”—def
Learning.feedbackCondAction.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [h_obl : IsObliviousEnv env] (n : β„•) :
  ProbabilityTheory.Kernel 𝓐 𝓨
Learning.feedbackCondAction.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [h_obl : IsObliviousEnv env] (n : β„•) :
  ProbabilityTheory.Kernel 𝓐 𝓨

Code

noncomputable
def feedbackCondAction (env : Environment 𝓐 𝓨) [h_obl : IsObliviousEnv env] (n : β„•) : Kernel 𝓐 𝓨 :=
  h_obl.exists_eq_prodMkLeft.choose n
Type uses (2)
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Dependency graph

Type dependencies (2)

EnvironmentπŸ”—

StructureLearning.Environment
Details

A stochastic environment.

πŸ”—structure
Learning.Environment.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : β„•) β†’ Kernel ((Iic n β†’ 𝓐 Γ— 𝓨) Γ— 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : βˆ€ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  Ξ½0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel Ξ½0]
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IsObliviousEnvπŸ”—

Type ClassLearning.IsObliviousEnv
Details

An environment is oblivious if the distribution of the next feedback depends only on the last action and not on the past history.

πŸ”—type class
Learning.IsObliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) : Prop
Learning.IsObliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) : Prop

Code

class IsObliviousEnv (env : Environment 𝓐 𝓨) : Prop where
  exists_eq_prodMkLeft : βˆƒ Ξ½ : β„• β†’ Kernel 𝓐 𝓨, (βˆ€ n, IsMarkovKernel (Ξ½ n)) ∧
    (env.Ξ½0 = Ξ½ 0) ∧ (βˆ€ n, env.feedback n = (Ξ½ (n + 1)).prodMkLeft _)
Type uses (1)
Used by (13)

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