Finite-spin random-scan heat-bath update

Input a finite nonempty site set $V$, a finite nonempty spin set $S$ with a fixed total order $s_1\prec \cdots \prec s_r$, a strictly positive finite weight $w:S^V\to (0,\mathrm{\infty })$, an initial configuration $x_0\in S^V$, an integer $t\ge 0$, a site schedule $i_1,\dots ,i_t\in V$, numbers $u_1,\dots ,u_t\in [0,1]$, and a finite family of observables $f_1,\dots ,f_m:S^V\to \mathbb{R}$ (the family may be empty). Put $Z={\sum }_{x\in S^V}w(x)$ and $\pi (x)=w(x)\mathrm{/}Z$. For $i\in V$, an exterior configuration $\eta \in S^{V\setminus \{i\}}$, and $s\in S$, let ${\eta }^{i,s}\in S^V$ agree with $\eta$ away from $i$ and have spin $s$ at $i$, and set $q_i(s\mid \eta )=\frac{w({\eta }^{i,s})}{{\sum }_{a\in S}w({\eta }^{i,a})}\mathrm{.}$ The denominator is finite and strictly positive. For $k=1,\dots ,t$, set $\eta =x_{k-1}{\mathrm{\mid }}_{V\setminus \{i_k\}}$, put $c_0=0$ and $c_j={\sum }_{\mathrm{\ell }=1}^j{q}_{i_k}(s_{\mathrm{\ell }}\mid \eta )$, and replace the spin at $i_k$ by the first $s_j$ for which $u_k<c_j$; when $u_k=1$, replace it by $s_r$. Leave every other spin unchanged to obtain $x_k$. Thus the intervals are $[c_{j-1},c_j)$ for $j<r$ and $[c_{r-1},1]$ for $j=r$. Return the trajectory $(x_0,\dots ,x_t)$ and the values $f_a(x_k)$ for $a=1,\dots ,m$ and $k=0,\dots ,t$. For the random-scan heat-bath chain, take the $i_k$ independently uniform on $V$ and the $u_k$ independently uniform on $[0,1]$, with all site and uniform draws mutually independent. The procedure makes no independence claim about successive configurations or observable values.

For a separately proved finite-time bias bound under explicit influence hypotheses, see Finite Dobrushin bound for random-scan heat-bath sampling. A finite run can still have initialization bias, and its outputs are generally dependent; this definition gives no confidence interval, independent-sample guarantee, or central limit theorem.