Prior-preserving Crank–Nicolson sampler

Assume the Axiom of Choice. Let $H$ be a separable real Hilbert space, let ${\mu }_0$ be a centered Borel Gaussian probability measure on $H$, let $D\subseteq H$ be Borel with ${\mu }_0(D)=1$, and let ${\mathrm{\Phi }}_D:D\to (-\mathrm{\infty },+\mathrm{\infty }]$ be Borel. Assume the evidence

$$Z:={\int }_D{e}^{-{\mathrm{\Phi }}_D(u)}{\mu }_0(du)$$

satisfies $0<Z<\mathrm{\infty }$. Extend the potential to all of $H$ by setting $\bar{\mathrm{\Phi }}(u)={\mathrm{\Phi }}_D(u)$ for $u\in D$ and $\bar{\mathrm{\Phi }}(u)=+\mathrm{\infty }$ for $u\mathrm{\notin }D$, and define the target probability $\mu (du)=Z^{-1}{e}^{-\bar{\mathrm{\Phi }}(u)}{\mu }_0(du)$. For a scale $\beta \in [0,1]$, write $\rho =\sqrt{1-{\beta }^2}$. At state $u\in H$, draw $\xi \sim {\mu }_0$ independently and propose $v=\rho u+\beta \xi$. Set the acceptance probability to $0$ if either $\bar{\mathrm{\Phi }}(u)$ or $\bar{\mathrm{\Phi }}(v)$ is $+\mathrm{\infty }$, and otherwise set

$$\alpha (u,v)=\min \{1,\exp (\bar{\mathrm{\Phi }}(u)-\bar{\mathrm{\Phi }}(v))\}\mathrm{.}$$

Draw an independent uniform random variable on $[0,1]$ and move to $v$ if it is at most $\alpha (u,v)$; otherwise remain at $u$. The resulting one-step transition rule is the prior-preserving Crank–Nicolson, or pCN, rule $P_{\beta }$. States with infinite potential are absorbing under this total definition; they have $\mu$-measure zero.

A prior on the declared Borel domain has full mass there, and its restriction is a probability measure.

A fixed-data potential defines evidence and a normalized target only when evidence is positive and finite.

A centered Borel Gaussian random element is Borel measurable and has centered normal continuous-linear projections.

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