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Learning.Ξ½0_obliviousEnvπŸ”—

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Ξ½0_obliviousEnvπŸ”—

LemmaLearning.Ξ½0_obliviousEnv
Details

No docstring.

πŸ”—theorem
Learning.Ξ½0_obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•), ProbabilityTheory.IsMarkovKernel (Ξ½ n)] :
  Environment.Ξ½0 (obliviousEnv Ξ½) = Ξ½ 0
Learning.Ξ½0_obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•),
      ProbabilityTheory.IsMarkovKernel
        (Ξ½ n)] :
  Environment.Ξ½0 (obliviousEnv Ξ½) = Ξ½ 0

Code

lemma Ξ½0_obliviousEnv (Ξ½ : β„• β†’ Kernel 𝓐 𝓨) [βˆ€ n, IsMarkovKernel (Ξ½ n)] :
    (obliviousEnv Ξ½).Ξ½0 = Ξ½ 0
Type uses (2)
Used by (1)

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Proof
by simp [obliviousEnv]

Dependency graph

Type dependencies (2)

obliviousEnvπŸ”—

DefinitionLearning.obliviousEnv
Details

An oblivious environment, in which the distribution of the next feedback depends only on the last action, but in a possibly time-dependent manner.

πŸ”—def
Learning.obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•), ProbabilityTheory.IsMarkovKernel (Ξ½ n)] : Environment 𝓐 𝓨
Learning.obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•),
      ProbabilityTheory.IsMarkovKernel
        (Ξ½ n)] :
  Environment 𝓐 𝓨

Code

def obliviousEnv (Ξ½ : β„• β†’ Kernel 𝓐 𝓨) [βˆ€ n, IsMarkovKernel (Ξ½ n)] : Environment 𝓐 𝓨 where
  feedback n := (Ξ½ (n + 1)).prodMkLeft _
  Ξ½0 := Ξ½ 0
Type uses (1)
Used by (10)

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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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