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Learning.IT.isBayesAlgEnvSeq_bayesTrajMeasure🔗

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Minimal Lean file

isBayesAlgEnvSeq_bayesTrajMeasure🔗

LemmaLearning.IT.isBayesAlgEnvSeq_bayesTrajMeasure
Details

No docstring.

🔗theorem
Learning.IT.isBayesAlgEnvSeq_bayesTrajMeasure.{u_1, u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) :
  IsBayesAlgEnvSeq Q κ alg (fun ω => Prod.fst (Prod.snd (ω 0))) action
    (fun n ω => Prod.snd (Prod.snd (ω n))) (bayesTrajMeasure Q κ alg)
Learning.IT.isBayesAlgEnvSeq_bayesTrajMeasure.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) :
  IsBayesAlgEnvSeq Q κ alg
    (fun ω => Prod.fst (Prod.snd (ω 0)))
    action
    (fun n ω => Prod.snd (Prod.snd (ω n)))
    (bayesTrajMeasure Q κ alg)

Code

lemma isBayesAlgEnvSeq_bayesTrajMeasure
    (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨) [IsMarkovKernel κ]
    (alg : Algorithm 𝓐 𝓨) :
    IsBayesAlgEnvSeq Q κ alg (fun ω ↦ (ω 0).2.1) action (fun n ω ↦ (ω n).2.2)
       (bayesTrajMeasure Q κ alg)
Type uses (5)
Body uses (7)
Used by (1)

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Proof
(isAlgEnvSeq_trajMeasure _ _).isBayesAlgEnvSeq

Dependency graph

Type dependencies (5)

Algorithm🔗

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

🔗structure
Learning.Algorithm.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
Used by (216)

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IsBayesAlgEnvSeq🔗

StructureLearning.IsBayesAlgEnvSeq
Details

IsBayesAlgEnvSeq Q κ alg E A Y P states that there is a measure P : Measure Ω such that the parameter E : Ω → 𝓔 has law Q and that the sequences of actions A : ℕ → Ω → 𝓐 and feedbacks Y : ℕ → Ω → 𝓨 are generated by the algorithm alg : Algorithm 𝓐 𝓨 interacting with an underlying environment that depends on E and κ (stationaryEnv (κ.sectR (E ω))).

🔗structure
Learning.IsBayesAlgEnvSeq.{u_1, u_2, u_3, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] [MeasurableSpace Ω]
  (Q : MeasureTheory.Measure 𝓔) (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  (alg : Algorithm 𝓐 𝓨) (E : Ω → 𝓔) (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
  (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsBayesAlgEnvSeq.{u_1, u_2, u_3,
    u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} {Ω : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] [MeasurableSpace Ω]
  (Q : MeasureTheory.Measure 𝓔)
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  (alg : Algorithm 𝓐 𝓨) (E : Ω → 𝓔)
  (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
  (P : MeasureTheory.Measure Ω)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsBayesAlgEnvSeq
    (Q : Measure 𝓔) (κ : Kernel (𝓔 × 𝓐) 𝓨) (alg : Algorithm 𝓐 𝓨)
    (E : Ω → 𝓔) (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] : Prop where
  measurable_param : Measurable E := by fun_prop
  measurable_action n : Measurable (A n) := by fun_prop
  measurable_feedback n : Measurable (Y n) := by fun_prop
  hasLaw_env : HasLaw E Q P
  hasCondDistrib_action_zero : HasCondDistrib (A 0) E (Kernel.const _ alg.p0) P
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (fun ω ↦ (E ω, A 0 ω)) κ P
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (fun ω ↦ (E ω, history A Y n ω))
      ((alg.policy n).prodMkLeft _) P
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, E ω, A (n + 1) ω))
      (κ.prodMkLeft _) P
Type uses (2)
Used by (22)

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action🔗

DefinitionLearning.IT.action
Details

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

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

Code

def action (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓐 := (h n).1
Used by (31)

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bayesTrajMeasure🔗

DefinitionLearning.IT.bayesTrajMeasure
Details

A measure P on a measurable space that carries random variables E, A, and Y such that IsBayesAlgEnvSeq Q κ alg E A Y P.

🔗def
Learning.IT.bayesTrajMeasure.{u_1, u_2, u_3} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓔 × 𝓨)
Learning.IT.bayesTrajMeasure.{u_1, u_2,
    u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓔 × 𝓨)

Code

noncomputable
def bayesTrajMeasure (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨)
    [IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) : Measure (ℕ → 𝓐 × 𝓔 × 𝓨) :=
  trajMeasure (alg.prodLeft 𝓔) (bayesStationaryEnv Q κ)
deriving IsProbabilityMeasure
Type uses (1)
Body uses (3)
Used by (5)

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instIsProbabilityMeasureForallNatProdBayesTrajMeasure🔗

InstanceLearning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure
Details

No docstring.

🔗theorem
Learning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (bayesTrajMeasure Q κ alg)
Learning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (bayesTrajMeasure Q κ alg)

Code

deriving IsProbabilityMeasure
Type uses (2)
Body uses (3)
Used by (4)

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Proof
deriving IsProbabilityMeasure

All dependencies, transitively (12)

history🔗

DefinitionLearning.history
Details

History of the algorithm-environment sequence up to time n.

🔗def
Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)
Used by (72)

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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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stepKernel🔗

DefinitionLearning.stepKernel
Details

Kernel describing the distribution of the next action-feedback pair given the history up to n.

🔗def
Learning.stepKernel.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)
Learning.stepKernel.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel
    (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)

Code

noncomputable
def stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=
  alg.policy n ⊗ₖ env.feedback n
deriving IsMarkovKernel
Type uses (2)
Used by (17)

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instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n)
Type uses (1)
Used by (14)

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Proof
alg.h_policy n

instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback🔗

InstanceLearning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Environment.feedback env n)
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.feedback env n)

Code

instance (env : Environment 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (env.feedback n)
Type uses (1)
Used by (5)

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Proof
env.h_feedback n

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (n : ℕ) : ProbabilityTheory.IsMarkovKernel (stepKernel alg env n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (stepKernel alg env n)

Code

deriving IsMarkovKernel
Type uses (3)
Body uses (2)
Used by (10)

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Proof
deriving IsMarkovKernel

trajMeasure🔗

DefinitionLearning.trajMeasure
Details

Measure on the sequence of actions and observations generated by the algorithm/environment.

🔗def
Learning.trajMeasure.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)
Learning.trajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)

Code

noncomputable
def trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
    Measure (ℕ → 𝓐 × 𝓨) :=
  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)
deriving IsProbabilityMeasure
Type uses (2)
Body uses (2)
Used by (19)

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instIsProbabilityMeasureP0🔗

InstanceLearning.instIsProbabilityMeasureP0
Details

No docstring.

🔗theorem
Learning.instIsProbabilityMeasureP0.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0
Type uses (1)
Used by (13)

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Proof
alg.hp0

prodLeft🔗

DefinitionLearning.Algorithm.prodLeft
Details

An algorithm with observations in 𝓧 × 𝓨 obtained from an algorithm with observations in 𝓨 by ignoring the 𝓧 component of each observation.

🔗def
Learning.Algorithm.prodLeft.{u_1, u_2, u_4} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (𝓧 : Type u_4) [MeasurableSpace 𝓧] (alg : Algorithm 𝓐 𝓨) :
  Algorithm 𝓐 (𝓧 × 𝓨)
Learning.Algorithm.prodLeft.{u_1, u_2,
    u_4}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (𝓧 : Type u_4)
  [MeasurableSpace 𝓧]
  (alg : Algorithm 𝓐 𝓨) :
  Algorithm 𝓐 (𝓧 × 𝓨)

Code

def Algorithm.prodLeft (𝓧 : Type*) [MeasurableSpace 𝓧] (alg : Algorithm 𝓐 𝓨) :
    Algorithm 𝓐 (𝓧 × 𝓨) where
  policy n := (alg.policy n).comap (fun h i ↦ ((h i).1, (h i).2.2)) (by fun_prop)
  p0 := alg.p0
Type uses (1)
Body uses (2)
Used by (6)

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bayesStationaryEnv🔗

DefinitionLearning.bayesStationaryEnv
Details

An environment with observations in 𝓔 × 𝓨. The first element e of an observation is sampled from Q once and remains constant. The second element of an observation is sampled from κ (e, a), where a is the corresponding action.

🔗def
Learning.bayesStationaryEnv.{u_1, u_2, u_3} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] : Environment 𝓐 (𝓔 × 𝓨)
Learning.bayesStationaryEnv.{u_1, u_2,
    u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] :
  Environment 𝓐 (𝓔 × 𝓨)

Code

noncomputable
def bayesStationaryEnv (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨)
    [IsMarkovKernel κ] : Environment 𝓐 (𝓔 × 𝓨) where
  feedback n :=
    let g : (Iic n → 𝓐 × 𝓔 × 𝓨) × 𝓐 → 𝓔 × 𝓐 := fun (h, a) => ((h ⟨0, by simp⟩).2.1, a)
    (Kernel.deterministic (Prod.fst ∘ g) (by fun_prop)) ×ₖ (κ.comap g (by fun_prop))
  ν0 := (Kernel.const _ Q) ⊗ₖ κ.swapLeft
Type uses (1)
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instIsMarkovKernelν0🔗

InstanceLearning.instIsMarkovKernelν0
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelν0.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel (Environment.ν0 env)
Learning.instIsMarkovKernelν0.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.ν0 env)

Code

instance (env : Environment 𝓐 𝓨) : IsMarkovKernel env.ν0
Type uses (1)
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Proof
env.hp0

instIsProbabilityMeasureForallNatProdTrajMeasure🔗

InstanceLearning.instIsProbabilityMeasureForallNatProdTrajMeasure
Details

No docstring.

🔗theorem
Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (trajMeasure alg env)
Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (trajMeasure alg env)

Code

deriving IsProbabilityMeasure
Type uses (3)
Body uses (4)
Used by (8)

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Proof
deriving IsProbabilityMeasure