`Learning.IsBayesAlgEnvSeq.hasLaw_history_withDensity`🔗

Lemma

No docstring.

theorem

Learning.IsBayesAlgEnvSeq.hasLaw_history_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}
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel κ] {Ω : Type u_4}
  [MeasurableSpace Ω] {E : Ω → 𝓔} {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  {alg : Algorithm 𝓐 𝓨} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {Ω₀ : Type u_5}
  [MeasurableSpace Ω₀] {E₀ : Ω₀ → 𝓔} {A₀ : ℕ → Ω₀ → 𝓐} {Y₀ : ℕ → Ω₀ → 𝓨}
  {alg₀ : Algorithm 𝓐 𝓨} {P₀ : MeasureTheory.Measure Ω₀}
  [MeasureTheory.IsProbabilityMeasure P₀]
  (h : IsBayesAlgEnvSeq Q κ alg E A Y P)
  (h₀ : IsBayesAlgEnvSeq Q κ alg₀ E₀ A₀ Y₀ P₀)
  (hc : Algorithm.AbsolutelyContinuous alg alg₀) (n : ℕ) :
  ProbabilityTheory.HasLaw (history A Y n)
    (MeasureTheory.Measure.withDensity
      (MeasureTheory.Measure.map (history A₀ Y₀ n) P₀)
      (Algorithm.density alg alg₀ n))
    P
Learning.IsBayesAlgEnvSeq.hasLaw_history_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}
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨}
  [ProbabilityTheory.IsMarkovKernel κ]
  {Ω : Type u_4} [MeasurableSpace Ω]
  {E : Ω → 𝓔} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {Ω₀ : Type u_5} [MeasurableSpace Ω₀]
  {E₀ : Ω₀ → 𝓔} {A₀ : ℕ → Ω₀ → 𝓐}
  {Y₀ : ℕ → Ω₀ → 𝓨} {alg₀ : Algorithm 𝓐 𝓨}
  {P₀ : MeasureTheory.Measure Ω₀}
  [MeasureTheory.IsProbabilityMeasure P₀]
  (h : IsBayesAlgEnvSeq Q κ alg E A Y P)
  (h₀ :
    IsBayesAlgEnvSeq Q κ alg₀ E₀ A₀ Y₀ P₀)
  (hc :
    Algorithm.AbsolutelyContinuous alg
      alg₀)
  (n : ℕ) :
  ProbabilityTheory.HasLaw (history A Y n)
    (MeasureTheory.Measure.withDensity
      (MeasureTheory.Measure.map
        (history A₀ Y₀ n) P₀)
      (Algorithm.density alg alg₀ n))
    P

Code

lemma hasLaw_history_withDensity (h : IsBayesAlgEnvSeq Q κ alg E A Y P)
    (h₀ : IsBayesAlgEnvSeq Q κ alg₀ E₀ A₀ Y₀ P₀) (hc : alg ≪ₐ alg₀) (n : ℕ) :
    HasLaw (history A Y n)
      ((P₀.map (history A₀ Y₀ n)).withDensity (alg.density alg₀ n)) P where
  aemeasurable

Proof

(measurable_history h.measurable_action h.measurable_feedback n).aemeasurable
  map_eq := by
    have hA := h.measurable_action
    have hY := h.measurable_feedback
    have hA₀ := h₀.measurable_action
    have hY₀ := h₀.measurable_feedback
    have hE := h.measurable_param
    have hE₀ := h₀.measurable_param
    rw [← condDistrib_comp_map hE.aemeasurable (by fun_prop), h.hasLaw_env.map_eq,
          Measure.bind_congr_right (h.condDistrib_history_eq_condDistrib_hist_withDensity h₀ hc n),
          Kernel.comp_withDensity_eq_withDensity_comp (by fun_prop),
          ← h₀.hasLaw_env.map_eq, condDistrib_comp_map hE₀.aemeasurable (by fun_prop)]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 6 project declarations, 96 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IsBayesAlgEnvSeq.hasLaw_history_withDensity🔗

`Learning.IsBayesAlgEnvSeq.hasLaw_history_withDensity`🔗