Exact Gibbs block update

Inputs: The finite product state space and target law from Proper full conditional kernel, a nonempty block $B$, its supplied proper full conditional kernel $K_B$, and a current state $x\in S$.

Output: A new state $x^{\mathrm{\prime }}\in S$ whose retained coordinates equal $x_{-B}$ and whose updated block is sampled from $K_B(x_{-B},\cdot )$.

The transition kernel is $P_B(x,C)={\int }_{S_B}{\mathbf{1}}_C\text{\hspace{0.17em}⁣}({\mathrm{splice}}_B(x_{-B},u))K_B(x_{-B},du),C\in \mathcal{S}\mathrm{.}$ Operationally, retain every coordinate outside $B$, draw $u$ from the full conditional kernel at the retained value, and replace the coordinates in $B$ by $u$. The finite product sigma-algebra makes the splice map measurable; the kernel-integral theorem makes the displayed evaluation measurable in $x$. The well-definedness result is The Gibbs block update is a probability kernel ↗.