Gibbs Sampler

Let

be a finite set of ``sites'' and $\Lambda$ be a finite set of ``common states.'' Consider a distribution $\pi$ on the space

$\displaystyle S := \{ (x_v) : (x_v) \in \Lambda^V \}$

of ``configurations.'' Then the Gibbs sampler runs as follows:

Suppose that at time .
Pick a site $v\in V$ , and generate $\lambda$ from the conditional distribution $\pi(x_v \vert (x_w)_{ w\neq v})$ .
Set $X_{t+1} = (x; x_v \gets \lambda)$ .

where $(x_w)_{ w\neq v}$ is restricted on $V\setminus\{v\}$ , and $(x; x_v \gets \lambda)$ replaces the state $x_v \gets \lambda$ .