AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath

- Resample coordinate `i` conditionally on all remaining coordinates. Only the selected coordinate must be nonempty and Standard Borel. The target may be any finite measure, including zero; its conditional version is specified only almost everywhere under the retained-coordinate marginal.

noncomputable def heatBath (μ : Measure (∀ j, X j)) [IsFiniteMeasure μ]
    (i : Fin (n + 1)) [StandardBorelSpace (X i)] [Nonempty (X i)] :
    Kernel (∀ j, X j) (∀ j, X j) :=
  let e := (MeasurableEquiv.piFinSuccAbove X i).trans MeasurableEquiv.prodComm
  ((HeatBath.heatBathSnd (μ.map e)).comap e e.measurable).map e.symm

/-- The coordinate update has total mass one at every input. This does not
assert conditional support on marginal-null fibers. -/

AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:25published source at 0e31a3cda412

- The coordinate update has total mass one at every input. This does not assert conditional support on marginal-null fibers.

instance heatBath_isMarkovKernel (μ : Measure (∀ j, X j)) [IsFiniteMeasure μ]
    (i : Fin (n + 1)) [StandardBorelSpace (X i)] [Nonempty (X i)] :
    IsMarkovKernel (heatBath X μ i) := by
  unfold heatBath
  exact Kernel.IsMarkovKernel.map _
    ((MeasurableEquiv.piFinSuccAbove X i).trans MeasurableEquiv.prodComm).symm.measurable

/-- Exact invariance by literal reuse of one-block heat-bath and measurable
kernel transport. No convergence from other initial laws is inferred. -/

AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:33published source at 0e31a3cda412

- Exact invariance by literal reuse of one-block heat-bath and measurable kernel transport. No convergence from other initial laws is inferred.

theorem heatBath_invariant (μ : Measure (∀ j, X j)) [IsFiniteMeasure μ]
    (i : Fin (n + 1)) [StandardBorelSpace (X i)] [Nonempty (X i)] :
    (heatBath X μ i).Invariant μ := by
  let e := (MeasurableEquiv.piFinSuccAbove X i).trans MeasurableEquiv.prodComm
  simpa only [heatBath, MeasurableEquiv.symm_symm, MeasurableEquiv.map_symm_map] using
    KernelTransport.invariant_map_comap e.symm (HeatBath.heatBathSnd_invariant (μ.map e))

/-- At each input `x`, every coordinate other than the selected one is retained
almost surely. Singleton measurability is needed only at the retained coordinate
`j`: equality of a marginal law with a Dirac measure on a coarse measurable
space would not alone imply literal almost-everywhere equality. -/

AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:42published source at 0e31a3cda412

- At each input `x`, every coordinate other than the selected one is retained almost surely. Singleton measurability is needed only at the retained coordinate `j`: equality of a marginal law with a Dirac measure on a coarse measurable space would not alone imply literal almost-everywhere equality.

theorem heatBath_ae_apply_eq (μ : Measure (∀ j, X j)) [IsFiniteMeasure μ]
    (i : Fin (n + 1)) [StandardBorelSpace (X i)] [Nonempty (X i)]
    (j : Fin (n + 1)) [MeasurableSingletonClass (X j)] (hji : j ≠ i)
    (x : ∀ j, X j) :
    ∀ᵐ y ∂heatBath X μ i x, y j = x j := by
  obtain ⟨k, rfl⟩ := Fin.exists_succAbove_eq hji
  let e := (MeasurableEquiv.piFinSuccAbove X i).trans MeasurableEquiv.prodComm
  let ν := condDistrib Prod.snd Prod.fst (μ.map e) (e x).1
  have hret : ∀ᵐ z ∂Measure.dirac (e x).1, z k = (e x).1 k :=
    ae_eq_dirac' (measurable_pi_apply k)
  have hfirst : ((Measure.dirac (e x).1).prod ν).map Prod.fst =
      Measure.dirac (e x).1 := by
    simp [ν]
  have hprod : ∀ᵐ p ∂(Measure.dirac (e x).1).prod ν, p.1 k = (e x).1 k :=
    ae_of_ae_map measurable_fst.aemeasurable (by rw [hfirst]; exact hret)
  have hcoord (p : (∀ j, X (i.succAbove j)) × X i) :
      e.symm p (i.succAbove k) = p.1 k :=
    congrArg (fun q => q.1 k) (e.apply_symm_apply p)
  change ∀ᵐ y ∂((HeatBath.heatBathSnd (μ.map e)).comap e e.measurable).map e.symm x,
    y (i.succAbove k) = x (i.succAbove k)
  rw [Kernel.map_apply _ e.symm.measurable, Kernel.comap_apply, HeatBath.heatBathSnd_apply]
  apply (ae_map_iff e.symm.measurable.aemeasurable
    ((measurable_pi_apply (i.succAbove k)) (measurableSet_singleton _))).2
  change ∀ᵐ p ∂(Measure.dirac (e x).1).prod ν,
    e.symm p (i.succAbove k) = (e x).1 k
  exact hprod.mono (fun p hp => (hcoord p).trans hp)

end AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath

AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:53published source at 0e31a3cda412