`ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure`🔗

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Minimal Lean file

`hasCondDistrib_trajMeasure`🔗

LemmaProbabilityTheory.Kernel.hasCondDistrib_trajMeasure

Details

No docstring.

theorem

ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Learning.Algorithm 𝓐 𝓨)
  (env : Learning.Environment 𝓐 𝓨) (n : ℕ) :
  HasCondDistrib (Learning.IT.step (n + 1)) (Learning.IT.hist n)
    (Learning.stepKernel alg env n) (Learning.trajMeasure alg env)
ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Learning.Algorithm 𝓐 𝓨)
  (env : Learning.Environment 𝓐 𝓨)
  (n : ℕ) :
  HasCondDistrib
    (Learning.IT.step (n + 1))
    (Learning.IT.hist n)
    (Learning.stepKernel alg env n)
    (Learning.trajMeasure alg env)

Code

lemma _root_.ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure
    (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    HasCondDistrib (step (n + 1)) (hist n) (stepKernel alg env n) (trajMeasure alg env)

Type uses (6)

Body uses (5)

Used by (1)

hasCondDistrib_step

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Proof

⟨by fun_prop, Kernel.map_frestrictLe_trajMeasure_compProd_eq_map_trajMeasure.symm⟩

Dependency graph

Type dependencies (6)

`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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`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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`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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`hist`🔗

DefinitionLearning.IT.hist

Details

hist n is the history up to time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

def

Learning.IT.hist.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.IT.hist.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} (n : ℕ) (h : ℕ → 𝓐 × 𝓨) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def hist (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : Iic n → 𝓐 × 𝓨 := fun i ↦ h i

Used by (23)

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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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`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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All dependencies, transitively (3)

`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)

Algorithm

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)

Environment

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

ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure🔗

hasCondDistrib_trajMeasure🔗

Algorithm🔗

Environment🔗

step🔗

hist🔗

stepKernel🔗

trajMeasure🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel🔗

`ProbabilityTheory.Kernel.hasCondDistrib_trajMeasure`🔗

`hasCondDistrib_trajMeasure`🔗

`Algorithm`🔗

`Environment`🔗

`step`🔗

`hist`🔗

`stepKernel`🔗

`trajMeasure`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel`🔗