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The pCN accept-reject procedure defines a probability kernel

Statement

Under the inputs and instructions of the prior-preserving Crank–Nicolson algorithm, for every $β \in [0, 1]$ its proposal rule is a Borel probability kernel and its accept-reject transition $P_{β}$ is a Borel probability kernel on $H$ .

Statement references

The proposal and accept-reject instructions, including the extended potential convention.

A probability kernel has probability-measure rows and measurable event evaluations.

Facts & Assumptions

Given: The inputs of Prior-preserving Crank–Nicolson sampler, a scale $β \in [0, 1]$ , the proposal $v = \sqrt{1 - β^{2}} u + β ξ$ , and the acceptance function specified there.

[F1]

The algorithm supplies Borel $D$ and $Φ_{D}$ , extends $Φ_{D}$ by $+ \infty$ off $D$ , proposes $ρ u + β ξ$ with independent $ξ \sim μ_{0}$ , and sets acceptance to zero when either potential value is infinite. Prior-preserving Crank–Nicolson sampler

[F2]

A probability kernel has probability-measure rows and measurable event evaluations. Measure kernel and probability kernel

[F3]

Integrating a nonnegative product-measurable function against a probability kernel gives a measurable function of the source point. Measurability of integration against a kernel

[F4]

For a measure $λ$ and nonnegative measurable function $f$ , the set function $A \mapsto \int_{A} f d λ$ is a measure. The indefinite integral of a nonnegative measurable function is a measure

[F5]

The preimage of a Borel set under a continuous map is Borel. A continuous map has Borel preimages of Borel sets

[F6]

Since $H$ is separable by the algorithm, let $D$ be a countable dense subset. The norm topology is the induced metric topology, whose positive-radius balls are open; the norm balls with centers in $D$ and positive rational radii are countable and form a base: if $x \in U$ with $U$ open, choose $r > 0$ with $B (x, r) \subseteq U$ , then choose $d \in D$ with $∥ d - x ∥ < r / 3$ and a rational $s$ with $∥ d - x ∥ < s < r - ∥ d - x ∥$ ; the triangle inequality gives $x \in B (d, s) \subseteq U$ . Thus $H$ is second-countable. The Borel sigma- algebra on $H \times H$ therefore equals $B (H) \otimes B (H)$ . Prior-preserving Crank–Nicolson sampler Separability: the existence of an at most countable dense subset Second countability: an at most countable basis for the topology A product of two at most countable sets is at most countable $Q$ is countably infinite The rationals embed densely in the reals A norm on a real vector space, the induced metric, and the dictionary with the metric axioms Open ball, closed ball and sphere in a metric space The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed Every subset of an at most countable set is at most countable Borel sigma-algebra of a product of second-countable spaces

[F7]

Composing a measurable function with a Borel scalar function preserves measurability; the real exponential is continuous by the supplied smoothness result, and finite arithmetic and lattice operations preserve measurability. Composition with a Borel measurable outer map preserves measurability The exponential function is strictly increasing Arithmetic and lattice operations preserve measurability whenever they are defined

Proof

Given: The hypotheses and algorithm inputs in the Statement.

1.1F1givenalgebra

For each Borel set $B$ in the extended potential range, the preimage ${\overset{ˉ}{Φ}}^{- 1} (B)$ is the union of $D \cap Φ_{D}^{- 1} (B)$ and $H ∖ D$ when $+ \infty \in B$ , and is just $D \cap Φ_{D}^{- 1} (B)$ otherwise. Since $D$ is Borel and $Φ_{D}$ is Borel on the trace Borel structure of $D$ , every such preimage is Borel in $H$ . Thus the extended potential used by the algorithm is Borel.

1.2F1F2F3F5F6givenconstruct

Write $ρ = \sqrt{1 - β^{2}}$ and define $Q_{β} (u, B) = \int_{H} 1_{B} (ρ u + β z) μ_{0} (d z)$ for Borel $B \subseteq H$ . For fixed $u$ , this is the pushforward of the probability $μ_{0}$ under the continuous map $z \mapsto ρ u + β z$ , so it is a probability measure in $B$ . The joint map $(u, z) \mapsto ρ u + β z$ is continuous; by [F6], its Borel preimages are product-measurable for $B (H) \otimes B (H)$ . Hence $(u, z) \mapsto 1_{B} (ρ u + β z)$ is product-measurable. The constant kernel $u \mapsto μ_{0}$ is a probability kernel, so [F3] makes $u \mapsto Q_{β} (u, B)$ measurable. Therefore $Q_{β}$ is a probability kernel by [F2].

2.1F1F2F3F5F6F7step 1.1step 1.2given

The finite-potential set $E = {u : \overset{ˉ}{Φ} (u) < + \infty}$ is Borel by step 1.1. The coordinate projections on $H \times H$ are continuous and hence Borel by [F5]; [F6] identifies their source Borel sigma-algebra with the product sigma-algebra. Composing them with the Borel $\overset{ˉ}{Φ}$ makes $(u, v) \mapsto \overset{ˉ}{Φ} (u)$ and $(u, v) \mapsto \overset{ˉ}{Φ} (v)$ Borel. On $E \times E$ their difference is real- valued and Borel, and [F7] makes its exponential Borel. Extending $\min {1, \exp (\overset{ˉ}{Φ} (u) - \overset{ˉ}{Φ} (v))}$ by $0$ off $E \times E$ therefore makes the algorithm's $α$ a Borel function with values in $[0, 1]$ . Put $a (u) = \int_{H} α (u, v) Q_{β} (u, d v)$ . By [F3], $a$ is measurable; the probability rows of $Q_{β}$ give $0 \leq a (u) \leq 1$ .

3.1F2F3F4step 2.1constructalgebra

For each $u$ and Borel $A \subseteq H$ , define $P_{β} (u, A) = \int_{A} α (u, v) Q_{β} (u, d v) + 1_{A} (u) (1 - a (u)) .$ The first term is a measure in $A$ by [F4] and step 2.1. The second term is also a measure: among any pairwise-disjoint sequence of sets, at most one contains $u$ , so the indicator of their union is the sum of their indicators at $u$ . Their sum has total mass $a (u) + 1 - a (u) = 1$ . For fixed $A$ , the first term is measurable in $u$ by [F3], and the second is measurable because both $1_{A} (u)$ and $a (u)$ are measurable. Thus $P_{β}$ is a probability kernel by [F2].

4.1step 1.1step 1.2step 2.1step 3.1∎

Steps 1.1, 1.2, 2.1, and 3.1 establish that the potential extension, the proposal, the acceptance function, and the accept-reject transition have the claimed Borel and probability properties.

Depends on

Used by

The pCN accept-reject kernel preserves the posterior Theorem

Cited to discharge well-definedness by Prior-preserving Crank–Nicolson sampler.

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Sources

S. L. Cotter, G. O. Roberts, A. M. Stuart, and D. White, MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster, Statistical Science 28(3) (2013) (standard reference, not scraped)
Yury Korolev and Jonas Latz, Inverse Problems: Lecture Notes, Michaelmas Term 2020 (University of Cambridge, 18 May 2021) (standard reference, not scraped)