`Learning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero`🔗

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Minimal Lean file

`hasCondDistrib_IT_feedback_zero`🔗

LemmaLearning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero

Details

No docstring.

theorem

Learning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero.{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]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐] [StandardBorelSpace 𝓨]
  [Nonempty 𝓨] [ProbabilityTheory.IsFiniteKernel κ]
  (h : IsBayesAlgEnvSeq Q κ alg E A Y P) :
  ∀ᵐ (e : 𝓔) ∂Q,
    ProbabilityTheory.HasCondDistrib (IT.feedback 0) (IT.action 0)
      (ProbabilityTheory.Kernel.sectR κ e) (𝓛[trajectory A Y | E; P] e)
Learning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero.{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]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  [ProbabilityTheory.IsFiniteKernel κ]
  (h : IsBayesAlgEnvSeq Q κ alg E A Y P) :
  ∀ᵐ (e : 𝓔) ∂Q,
    ProbabilityTheory.HasCondDistrib
      (IT.feedback 0) (IT.action 0)
      (ProbabilityTheory.Kernel.sectR κ e)
      (𝓛[trajectory A Y | E; P] e)

Code

lemma hasCondDistrib_IT_feedback_zero [IsFiniteKernel κ] (h : IsBayesAlgEnvSeq Q κ alg E A Y P) :
    ∀ᵐ e ∂Q, HasCondDistrib (IT.feedback 0) (IT.action 0) (κ.sectR e)
      (condDistrib (trajectory A Y) E P e)

Type uses (5)

Body uses (4)

Used by (1)

ae_IsAlgEnvSeq

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Proof

by
  rw [← h.hasLaw_env.map_eq]
  exact h.hasCondDistrib_feedback_zero.hasCondDistrib_sectR
    (IT.measurable_action 0) (IT.measurable_feedback 0)
    (measurable_trajectory h.measurable_action h.measurable_feedback).aemeasurable

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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`feedback`🔗

DefinitionLearning.IT.feedback

Details

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

def

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

Code

def feedback (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓨 := (h n).2

Used by (16)

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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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`trajectory`🔗

DefinitionLearning.trajectory

Details

A random variable that gives the sequence of action-feedback pairs.

def

Learning.trajectory.{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) (ω : Ω) : ℕ → 𝓐 × 𝓨
Learning.trajectory.{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) (ω : Ω) : ℕ → 𝓐 × 𝓨

Code

def trajectory (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (ω : Ω) : ℕ → 𝓐 × 𝓨 := fun n ↦ (A n ω, Y n ω)

Used by (18)

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All dependencies, transitively (1)

`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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Learning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero🔗

hasCondDistrib_IT_feedback_zero🔗

Algorithm🔗

IsBayesAlgEnvSeq🔗

feedback🔗

action🔗

trajectory🔗

history🔗

`Learning.IsBayesAlgEnvSeq.hasCondDistrib_IT_feedback_zero`🔗

`hasCondDistrib_IT_feedback_zero`🔗

`Algorithm`🔗

`IsBayesAlgEnvSeq`🔗

`feedback`🔗

`action`🔗

`trajectory`🔗

`history`🔗