A normalized mixture of Markov kernels is a Markov kernel

AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_isMarkovKernel · theorem · Teaching coverage

Statement

Let I be a finite index type, A and B measurable spaces, and Kᵢ Markov kernels from A to B. Let wᵢ be fixed nonnegative real weights whose sum is one. Then the finite mixture M_w is a Markov kernel: for every input a, its output measure is a probability measure. Individual weights may be zero.

All objects and hypotheses

• {ι α β : Type*}, [Fintype ι], [MeasurableSpace α], [MeasurableSpace β] from the local section.
• w : ι → ℝ≥0, independent of current state.
• κ : ι → Kernel α β with [∀ i, IsMarkovKernel (κ i)].
• hsum : ∑ i, w i = 1 as an equality in ℝ≥0.
• There is no separate IsSFiniteKernel argument: the Markov-kernel typeclass instances supply the construction contract automatically.
• No target measure, finiteness of state spaces, strict positivity of weights, nonempty-state assumption, reversibility or convergence assumption.

Notation and interpretation

Measurable space and kernel

A measurable space is a set with a specified sigma-algebra of events. A kernel K from A to B assigns to each input a a measure K(a,·) on B, and a↦K(a,E) is measurable whenever E is measurable. A kernel need not assign probability measures unless Markovness is separately assumed.

\[K:A\to\mathcal M(B),\qquad a\mapsto K(a,E)\text{ measurable for every measurable }E\subseteq B.\]

Measure evolved by a kernel; invariance

Write μK for the measure obtained by starting with μ and applying K. Lean writes μ.bind K or K ∘ₘ μ. Invariance says that this evolved measure is exactly μ. If μ is a probability law this means a stationary initial distribution, but the definitions and these theorems also allow arbitrary measures.

\[(\mu K)(E)=\int_A K(a,E)\,\mu(da),\qquad \mu K=\mu.\]

Composition and powers

Lean's L ∘ₖ K applies K first and L second. The multiplicative identity is the identity kernel a↦δ_a, and powers are repeated composition. All integrals here are nonnegative extended-real (lower Lebesgue) integrals; no signed or Bochner integrability is silently assumed.

\[(L\circ_{\!k}K)(a,E)=\int L(b,E)\,K(a,db),\quad K^0(a,\cdot)=\delta_a,\quad K^{n+1}=K^n\circ_{\!k}K.\]

Markov and s-finite kernels

Markov means every output measure has total mass one. An s-finite kernel is a countable sum of finite kernels, where each finite kernel has a finite uniform bound on its total mass across inputs. Markovness supplies this s-finiteness automatically. S-finite is not the same hypothesis as sigma-finite, and is not merely a pointwise finiteness assertion.

\[K(a,B)=1\ \forall a\quad\text{(Markov)};\qquad K=\sum_{n=0}^{\infty}K_n,\quad \forall n\ \exists C_n<\infty\ \forall a,\ K_n(a,B)\le C_n\quad\text{(s-finite)}.\]

Nonnegative weights

ℝ≥0 consists of finite nonnegative real numbers; ℝ≥0∞ also allows infinity and is the codomain of measure values. The coercion wᵢ↦cᵢ puts finite weights into extended nonnegative arithmetic. Zero weights are permitted, including when component masses are infinite; this arithmetic uses 0·∞=0.

\[w_i\in[0,\infty),\qquad c_i=\iota(w_i)\in[0,\infty].\]

Measurable equivalence and pushforward

A measurable equivalence e is a bijection whose forward and inverse maps are both measurable. Its pushforward measure records the original mass of inverse images. This is transport of measures, not a density transformation with an implicit Jacobian.

\[(e_*\mu)(E)=\mu(e^{-1}(E)),\qquad (e^{-1})_*(e_*\mu)=\mu.\]

Mathematical proof

have hsum' : ∑ i, (w i : ℝ≥0∞) = 1 := by
  simpa using congrArg (fun r : ℝ≥0 => (r : ℝ≥0∞)) hsum

constructor
intro a
constructor

rw [finiteMixture_apply, Measure.finsetSum_apply]
simpa only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one] using hsum'

theorem finiteMixture_isMarkovKernel (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsMarkovKernel (κ i)] (hsum : ∑ i, w i = 1) :
    IsMarkovKernel (finiteMixture w κ)

theorem finiteMixture_isMarkovKernel (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsMarkovKernel (κ i)] (hsum : ∑ i, w i = 1) :
    IsMarkovKernel (finiteMixture w κ) := by
  have hsum' : ∑ i, (w i : ℝ≥0∞) = 1 := by
    simpa using congrArg (fun r : ℝ≥0 => (r : ℝ≥0∞)) hsum
  constructor
  intro a
  constructor
  rw [finiteMixture_apply, Measure.finsetSum_apply]
  simpa only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one] using hsum'

/-- A fixed normalized finite mixture preserves any common invariant measure.

There is no finite or s-finite assumption on `μ`. The component s-finiteness is
only the `withDensity` construction contract; intended Markov components satisfy
it automatically. This statement does not extend to state-dependent weights. -/

Scope and omitted-condition boundaries

• Normalization alone says nothing about a common invariant measure; finiteMixture_invariant addresses that separate assertion.
• Weights need not be positive. A mixture containing the identity kernel need not have positive holding probability.
• Empty I cannot satisfy hsum.
• No sampling implementation, state-dependent scan rule, reversibility, aperiodicity, convergence or mixing bound.
• The abstract normalized-mixture theorem is source-neutral; no numbered textbook theorem is assigned.