Random and systematic Gibbs scan schemes

Inputs: A common finite-product measurable state space $(S,\mathcal{S})$ with target probability $\pi$, an integer $m\ge 1$, and for each $j=1,\dots ,m$ a nonempty block $B_j$ with its supplied proper full conditional kernel for $\pi$. Invoke Exact Gibbs block update for each block to obtain its exact update kernel $P_j:S\to S$. Select random scan, systematic scan, or both; if random scan is selected, also supply fixed weights $w_j\ge 0$ with ${\sum }_{j=1}^m{w}_j=1$.

Admissible conditions: Every $P_j$ acts on this same $(S,\mathcal{S})$ and is an exact block-update kernel for this same target $\pi$. The weights are required only for the random-scan branch; systematic scan takes no weights.

Outputs: For each selected mode, its one-update random-scan kernel $P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}:S\to S$ or full-sweep systematic kernel $P_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{p}}:S\to S$; for systematic scan, also the phase-indexed single-update kernels $P_1,\dots ,P_m$.

Random-scan instructions: At each update, independently draw $j\in \{1,\dots ,m\}$ with probabilities $w_1,\dots ,w_m$ and apply $P_j$ to the current state. The one-update kernel is $P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}(x,A)={\sum }_{j=1}^m{w}_j{P}_j(x,A)\mathrm{.}$ Uniform random scan is the special case $w_j=1\mathrm{/}m$.

Systematic-scan instructions: Starting from the current state, apply $P_1$, then $P_2$, through $P_m$ in that fixed order; repeat the same order for each subsequent sweep. For kernels, $P_1\cdots P_m$ means chronological composition: apply $P_1$ first and $P_m$ last, as in Composition of probability kernels. Return the full-sweep kernel $P_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{p}}=P_1\cdots P_m$. Observing after each single update gives the phase-indexed sequence $P_1,\dots ,P_m,P_1,\dots$ and is generally time-inhomogeneous; observing only at full-sweep boundaries gives a homogeneous chain with kernel $P_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{p}}$. The scan kernels are well-defined by Random and systematic scan kernels are well-defined ↗.