MCMC

Dynamics of Gibbs Models

Suppose that $\Lambda$ consists of finite common states. For example, $\Lambda = \{-1,+1\}$ as in Ising model. Then we can construct the transition probability at each $v\in V$ by

$\displaystyle \mathbf{P}_v(x,y) := \begin{cases} \dfrac{\pi(y)}{\sum_{\lambda\... ...eq v} = (y_w)_{ w\neq v}$; } [0.5in] 0 & \mbox{ otherwise. } \end{cases}$

Site update. $\mathbf{P}_v$ is a reversible Markov chain with the stationary distribution $\pi$ since

$\pi(\mathbf{x}) \mathbf{P}_v(\mathbf{x},\mathbf{y}) = \pi(\mathbf{x}) \dfrac{\pi(\mathbf{y})} {\sum_{\lambda\in\Lambda} \pi((\mathbf{x}; x_v \gets \lambda))}$ $= \pi(\mathbf{y}) \dfrac{\pi(\mathbf{x})} {\sum_{\lambda\in\Lambda} \pi((\mat... ...y}; y_v \gets \lambda))} = \pi(\mathbf{y}) \mathbf{P}_v(\mathbf{y},\mathbf{x})$ .

However, it is not irreducible. Now take a fixed ordering $v_1,\ldots,v_n$ of . Define $\mathbf{P} := \mathbf{P}_{v_1} \cdots \mathbf{P}_{v_n}$ , call it systematic site update. Then $\mathbf{P}$ is an irreducible transition probability with the stationary distribution $\pi$ . Alternatively we can introduce a distribution $\rho(v)$ over , and define $\mathbf{P} = \sum_{v\in V} \rho(v) \mathbf{P}_{v}$ , which is called random site update.