LeanMachineLearning exposition

Learning.IsBayesAlgEnvSeq.condDistrib_history_eq_condDistrib_hist_withDensityπŸ”—

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condDistrib_history_eq_condDistrib_hist_withDensityπŸ”—

LemmaLearning.IsBayesAlgEnvSeq.condDistrib_history_eq_condDistrib_hist_withDensity
Details

No docstring.

πŸ”—theorem
Learning.IsBayesAlgEnvSeq.condDistrib_history_eq_condDistrib_hist_withDensity.{u_1,
    u_2, u_3, u_4, u_5}
  {𝓐 : Type u_1} {𝓨 : Type u_2} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  {𝓔 : Type u_3} [MeasurableSpace 𝓔] [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨] {Q : MeasureTheory.Measure 𝓔}
  {ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel ΞΊ] {Ξ© : Type u_4}
  [MeasurableSpace Ξ©] {E : Ξ© β†’ 𝓔} {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  {alg : Algorithm 𝓐 𝓨} {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P] {Ξ©β‚€ : Type u_5}
  [MeasurableSpace Ξ©β‚€] {Eβ‚€ : Ξ©β‚€ β†’ 𝓔} {Aβ‚€ : β„• β†’ Ξ©β‚€ β†’ 𝓐} {Yβ‚€ : β„• β†’ Ξ©β‚€ β†’ 𝓨}
  {algβ‚€ : Algorithm 𝓐 𝓨} {Pβ‚€ : MeasureTheory.Measure Ξ©β‚€}
  [MeasureTheory.IsProbabilityMeasure Pβ‚€]
  (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P)
  (hβ‚€ : IsBayesAlgEnvSeq Q ΞΊ algβ‚€ Eβ‚€ Aβ‚€ Yβ‚€ Pβ‚€)
  (hc : Algorithm.AbsolutelyContinuous alg algβ‚€) (n : β„•) :
  ⇑𝓛[history A Y n | E; P] =ᡐ[Q]
    ⇑(ProbabilityTheory.Kernel.withDensity 𝓛[history Aβ‚€ Yβ‚€ n | Eβ‚€; Pβ‚€]
        fun x => Algorithm.density alg algβ‚€ n)
Learning.IsBayesAlgEnvSeq.condDistrib_history_eq_condDistrib_hist_withDensity.{u_1,
    u_2, u_3, u_4, u_5}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  {𝓔 : Type u_3} [MeasurableSpace 𝓔]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  {Q : MeasureTheory.Measure 𝓔}
  {ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel ΞΊ]
  {Ξ© : Type u_4} [MeasurableSpace Ξ©]
  {E : Ξ© β†’ 𝓔} {A : β„• β†’ Ξ© β†’ 𝓐}
  {Y : β„• β†’ Ξ© β†’ 𝓨} {alg : Algorithm 𝓐 𝓨}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P]
  {Ξ©β‚€ : Type u_5} [MeasurableSpace Ξ©β‚€]
  {Eβ‚€ : Ξ©β‚€ β†’ 𝓔} {Aβ‚€ : β„• β†’ Ξ©β‚€ β†’ 𝓐}
  {Yβ‚€ : β„• β†’ Ξ©β‚€ β†’ 𝓨} {algβ‚€ : Algorithm 𝓐 𝓨}
  {Pβ‚€ : MeasureTheory.Measure Ξ©β‚€}
  [MeasureTheory.IsProbabilityMeasure Pβ‚€]
  (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P)
  (hβ‚€ :
    IsBayesAlgEnvSeq Q ΞΊ algβ‚€ Eβ‚€ Aβ‚€ Yβ‚€ Pβ‚€)
  (hc :
    Algorithm.AbsolutelyContinuous alg
      algβ‚€)
  (n : β„•) :
  ⇑𝓛[history A Y n | E; P] =ᡐ[Q]
    ⇑(ProbabilityTheory.Kernel.withDensity
        𝓛[history Aβ‚€ Yβ‚€ n | Eβ‚€; Pβ‚€]
        fun x =>
        Algorithm.density alg algβ‚€ n)

Code

lemma condDistrib_history_eq_condDistrib_hist_withDensity (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P)
    (hβ‚€ : IsBayesAlgEnvSeq Q ΞΊ algβ‚€ Eβ‚€ Aβ‚€ Yβ‚€ Pβ‚€) (hc : alg β‰ͺₐ algβ‚€) (n : β„•) :
    condDistrib (history A Y n) E P =ᡐ[Q]
      ((condDistrib (history Aβ‚€ Yβ‚€ n) Eβ‚€ Pβ‚€).withDensity
        (fun _ ↦ alg.density algβ‚€ n))
Type uses (5)
Body uses (10)
Used by (2)

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Proof
by
    filter_upwards [h.ae_IsAlgEnvSeq, hβ‚€.ae_IsAlgEnvSeq, h.hasLaw_IT_hist n, hβ‚€.hasLaw_IT_hist n]
      with _ hae haeβ‚€ he heβ‚€
    rw [Kernel.withDensity_apply _ (by fun_prop), ← he.map_eq, ← heβ‚€.map_eq]
    exact (hae.hasLaw_history_withDensity haeβ‚€ hc n).map_eq

Dependency graph

Type dependencies (5)

AlgorithmπŸ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

πŸ”—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 : β„•) β†’ Kernel (Iic n β†’ 𝓐 Γ— 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : βˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
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IsBayesAlgEnvSeqπŸ”—

StructureLearning.IsBayesAlgEnvSeq
Details

IsBayesAlgEnvSeq Q ΞΊ alg E A Y P states that there is a measure P : Measure Ξ© such that the parameter E : Ξ© β†’ 𝓔 has law Q and that the sequences of actions A : β„• β†’ Ξ© β†’ 𝓐 and feedbacks Y : β„• β†’ Ξ© β†’ 𝓨 are generated by the algorithm alg : Algorithm 𝓐 𝓨 interacting with an underlying environment that depends on E and ΞΊ (stationaryEnv (ΞΊ.sectR (E Ο‰))).

πŸ”—structure
Learning.IsBayesAlgEnvSeq.{u_1, u_2, u_3, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} {Ξ© : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] [MeasurableSpace Ξ©]
  (Q : MeasureTheory.Measure 𝓔) (ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨)
  (alg : Algorithm 𝓐 𝓨) (E : Ξ© β†’ 𝓔) (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨)
  (P : MeasureTheory.Measure Ξ©) [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} {Ξ© : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] [MeasurableSpace Ξ©]
  (Q : MeasureTheory.Measure 𝓔)
  (ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨)
  (alg : Algorithm 𝓐 𝓨) (E : Ξ© β†’ 𝓔)
  (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨)
  (P : MeasureTheory.Measure Ξ©)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsBayesAlgEnvSeq
    (Q : Measure 𝓔) (ΞΊ : Kernel (𝓔 Γ— 𝓐) 𝓨) (alg : Algorithm 𝓐 𝓨)
    (E : Ξ© β†’ 𝓔) (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨)
    (P : Measure Ξ©) [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 Ο‰ ↦ (E Ο‰, A 0 Ο‰)) ΞΊ P
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (fun Ο‰ ↦ (E Ο‰, history A Y n Ο‰))
      ((alg.policy n).prodMkLeft _) P
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun Ο‰ ↦ (history A Y n Ο‰, E Ο‰, A (n + 1) Ο‰))
      (ΞΊ.prodMkLeft _) P
Type uses (2)
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AbsolutelyContinuousπŸ”—

StructureLearning.Algorithm.AbsolutelyContinuous
Details

For every time and history, the distribution over actions according to alg is absolutely continuous with respect to the distribution over actions according to algβ‚€.

πŸ”—structure
Learning.Algorithm.AbsolutelyContinuous.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (alg algβ‚€ : Algorithm 𝓐 𝓨) : Prop
Learning.Algorithm.AbsolutelyContinuous.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (alg algβ‚€ : Algorithm 𝓐 𝓨) : Prop

Code

structure AbsolutelyContinuous (alg algβ‚€ : Algorithm 𝓐 𝓨) : Prop where
  p0 : alg.p0 β‰ͺ algβ‚€.p0
  policy n h : alg.policy n h β‰ͺ algβ‚€.policy n h
Type uses (1)
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historyπŸ”—

DefinitionLearning.history
Details

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

πŸ”—def
Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐)
  (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨

Code

def history (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) : Iic n β†’ 𝓐 Γ— 𝓨 :=
  fun i ↦ (A i Ο‰, Y i Ο‰)
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densityπŸ”—

DefinitionLearning.Algorithm.density
Details

If the algorithm alg is absolutely continuous with respect to the algorithm algβ‚€ and they are both interacting with the same environment, then the law of the history at time n under alg is the law of the history at time n under algβ‚€ with density alg.density algβ‚€ n.

πŸ”—def
Learning.Algorithm.density.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  [MeasurableSpace.CountablyGenerated 𝓐] (alg algβ‚€ : Algorithm 𝓐 𝓨)
  (n : β„•) : (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) β†’ ENNReal
Learning.Algorithm.density.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  [MeasurableSpace.CountablyGenerated 𝓐]
  (alg algβ‚€ : Algorithm 𝓐 𝓨) (n : β„•) :
  (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) β†’ ENNReal

Code

noncomputable
def density [MeasurableSpace.CountablyGenerated 𝓐] (alg algβ‚€ : Algorithm 𝓐 𝓨) :
    (n : β„•) β†’ (Iic n β†’ 𝓐 Γ— 𝓨) β†’ ℝβ‰₯0∞
  | 0, h => (alg.p0.rnDeriv algβ‚€.p0 (h ⟨0, by simp⟩).1)
  | n + 1, h =>
    let p := MeasurableEquiv.IicSuccProd (fun _ ↦ 𝓐 Γ— 𝓨) n h
    alg.density algβ‚€ n p.1 * (alg.policy n).rnDeriv (algβ‚€.policy n) p.1 p.2.1
Type uses (1)
Body uses (1)
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All dependencies, transitively (1)

IicSuccProdπŸ”—

DefinitionMeasurableEquiv.IicSuccProd
Details

Measurable equivalence between a product up to n + 1 and the pair of the product up to n and the space at n + 1.

πŸ”—def
MeasurableEquiv.IicSuccProd.{u_3} (X : β„• β†’ Type u_3)
  [(n : β„•) β†’ MeasurableSpace (X n)] (n : β„•) :
  ((i : β†₯(Finset.Iic (n + 1))) β†’ X ↑i) ≃ᡐ
    ((i : β†₯(Finset.Iic n)) β†’ X ↑i) Γ— X (n + 1)
MeasurableEquiv.IicSuccProd.{u_3}
  (X : β„• β†’ Type u_3)
  [(n : β„•) β†’ MeasurableSpace (X n)]
  (n : β„•) :
  ((i : β†₯(Finset.Iic (n + 1))) β†’ X ↑i) ≃ᡐ
    ((i : β†₯(Finset.Iic n)) β†’ X ↑i) Γ—
      X (n + 1)

Code

def _root_.MeasurableEquiv.IicSuccProd (X : β„• β†’ Type*) [βˆ€ n, MeasurableSpace (X n)] (n : β„•) :
    MeasurableEquiv (Ξ  i : Iic (n + 1), X i) ((Ξ  i : Iic n, X i) Γ— X (n + 1)) :=
  (MeasurableEquiv.IicProdIoc (Nat.le_succ n)).symm.trans
    (MeasurableEquiv.prodCongr (MeasurableEquiv.refl _) (MeasurableEquiv.piSingleton n).symm)
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