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    • Gibbs Distribution

    Let $ G = (V,E)$ be a finite undirected graph. The site $ v$ is said to be a neighbor of $ w$ if $ \{v,w\}\in E$ . $ C \subset V$ is called a clique if $ \{v,w\}\in E$ whenever $ v,w\in C$ . We denote the set of all cliques by $ \mathcal{C}$ . Let $ \Lambda := \{1,\ldots,L\}$ be a common state space. Given a parameter $ \theta > 0$ , we can define a Gibbs distribution with respect to $ G$ by

    $\displaystyle \pi(x) := \frac{1}{z_\theta} \exp(-\theta H(x)), \quad x\in S $

    where

    $\displaystyle z_\theta = \sum_{x\in S} \exp(-\theta H(x)) $

    is the normalizing constant.

    Hamiltonian. In the language of statistical mechanics, a vertex $ v\in V$ is called a site. The parameter $ \theta$ and $ z_\theta$ are respectively called a ``inverse temperature'' and ``partition function.'' And $ H(x)$ , called a ``Hamiltonian,'' has the form

    $\displaystyle H(x) = \sum_{C \in \mathcal{C}} W_C(x), $

    where each $ W_C$ depends only on those coordinates $ x_v$ , $ v\in C$ . The Hamiltonian $ H(x)$ is a energy function, and the model abhors to retain a high energy when the temperature $ T = 1/\theta$ is low.

    Markov random field. Let $ X = (X_v)$ be a $ S$ -valued random variable. $ X$ is an MRF (Markov random field) with respect to $ G$ if for every $ v\in V$ and $ \mathbf{x}\in\mathcal{S}$

    1. $ P(X = x) > 0$ ;
    2. $ P(X_v = x_v \vert X_w = x_w, w \neq v )$

      $ = {P(X_v = x_v \vert X_w = x_w, \{w,v\}\in E )}$ .

    Hammersley-Clifford theorem. $ \mathbf{X}$ is an MRF with respect to $ G$ if and only if $ \pi(\mathbf{x}) = P(\mathbf{X} = \mathbf{x})$ is a Gibbs distribution with respect to $ G$ . In this sense a Gibbs distribution is also known as GRF (Gibbs random field).

    Site update. In the Gibbs distribution, the conditional probability for site update is given by

    $ \pi(x_v \vert (x_w)_{ w\neq v})$ $ = \dfrac{\exp(-\theta H_v(y))} {\sum_{\lambda\in\Lambda} \exp(-\theta H_v(x; x_v\gets\lambda))}$

    Here the normalizing constant $ z_\theta$ is canceled and

    $\displaystyle H_v(x) = \displaystyle\sum_{v\in C, C\in\mathcal{C}} W_C(x) $

    is only the partial sum over neighboring cliques of $ v$ .

    © TTU Mathematics