Constructing a finite fixed-weight mixture of kernels

AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture · def · Teaching coverage

Statement

Let I be a finite index type, A and B measurable spaces, w:I→[0,∞) fixed nonnegative real weights, and Kᵢ kernels from A to B. Assume every Kᵢ is an s-finite kernel. The construction first multiplies each output measure of Kᵢ by the constant density wᵢ, then adds these finitely many kernels. Denote the result by M_w. No sum-to-one condition is needed to define it; the next theorem identifies its value at each input as the weighted sum of component measures.

All objects and hypotheses

• Implicit section variables {ι α β : Type*}.
• [Fintype ι]: there are finitely many component indices; this does not make α or β a finite state space.
• [MeasurableSpace α] and [MeasurableSpace β].
• w : ι → ℝ≥0, fixed independently of input and output states.
• κ : ι → Kernel α β.
• [∀ i, IsSFiniteKernel (κ i)], a uniform-kernel decomposition contract for each component, required by the chosen withDensity API.
• No normalization, strict positivity, Markovness, target measure μ, nonempty-space, Standard-Borel or topological 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.

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

Construction and meaning

noncomputable def finiteMixture (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsSFiniteKernel (κ i)] : Kernel α β

noncomputable def finiteMixture (w : ι → ℝ≥0) (κ : ι → Kernel α β)
    [∀ i, IsSFiniteKernel (κ i)] : Kernel α β :=
  ∑ i, (κ i).withDensity (fun _ _ => (w i : ℝ≥0∞))

/-- Pointwise measure law of the finite mixture. Constant uncurry densities are
measurable, so the totalized `withDensity` zero fallback is never used. -/

Scope and omitted-condition boundaries

• This definition alone claims neither Markovness nor invariance.
• S-finiteness is the selected Mathlib implementation's contract; this is not a claim that every conceivable finite-mixture definition mathematically needs it.
• State-dependent weights, signed weights and computational sampling code are not provided.
• Empty I yields the zero kernel; the later hypothesis ∑ᵢwᵢ=1 cannot hold for empty I.
• Source-neutral construction; no numbered primary textbook theorem is assigned.