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Gibbs Sampler

Let $ V$ 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:
  1. Suppose that $ X_t = x$ at time $ t$.
  2. Pick a site $ v\in V$, and generate $ \lambda$ from the conditional distribution $ \pi(x_v \vert (x_w)_{ w\neq v})$.
  3. 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$.


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