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Learning.feedback_eq_feedbackCondAction🔗

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Minimal Lean file

feedback_eq_feedbackCondAction🔗

LemmaLearning.feedback_eq_feedbackCondAction
Details

No docstring.

🔗theorem
Learning.feedback_eq_feedbackCondAction.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [IsObliviousEnv env] (n : ℕ) :
  Environment.feedback env n =
    ProbabilityTheory.Kernel.prodMkLeft (↥(Finset.Iic n) → 𝓐 × 𝓨)
      (feedbackCondAction env (n + 1))
Learning.feedback_eq_feedbackCondAction.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [IsObliviousEnv env] (n : ℕ) :
  Environment.feedback env n =
    ProbabilityTheory.Kernel.prodMkLeft
      (↥(Finset.Iic n) → 𝓐 × 𝓨)
      (feedbackCondAction env (n + 1))

Code

lemma feedback_eq_feedbackCondAction (env : Environment 𝓐 𝓨) [IsObliviousEnv env] (n : ℕ) :
    env.feedback n = (feedbackCondAction env (n + 1)).prodMkLeft _
Type uses (3)
Used by (3)

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Proof
IsObliviousEnv.exists_eq_prodMkLeft.choose_spec.2.2 n

Dependency graph

Type dependencies (3)

Environment🔗

StructureLearning.Environment
Details

A stochastic environment.

🔗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 : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]
Used by (128)

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IsObliviousEnv🔗

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.

🔗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 : ∃ ν : ℕ → Kernel 𝓐 𝓨, (∀ n, IsMarkovKernel (ν n)) ∧
    (env.ν0 = ν 0) ∧ (∀ n, env.feedback n = (ν (n + 1)).prodMkLeft _)
Type uses (1)
Used by (13)

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feedbackCondAction🔗

DefinitionLearning.feedbackCondAction
Details

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

🔗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 : ℕ) :
  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 : ℕ) :
  ProbabilityTheory.Kernel 𝓐 𝓨

Code

noncomputable
def feedbackCondAction (env : Environment 𝓐 𝓨) [h_obl : IsObliviousEnv env] (n : ℕ) : Kernel 𝓐 𝓨 :=
  h_obl.exists_eq_prodMkLeft.choose n
Type uses (2)
Used by (12)

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