A Polynomial-time Perfect Sampler for the Q-Ising with local fields (Mathematical Foundation of Algorithms and Computer Science)

We present a polynomial-time perfect sampler for the Q-Ising with a vertex-independent noise. The Q-Ising, one of the generalized models of the Ising, arose in the context of Bayesian image restoration in statistical mechanics. We study the distribution of Q-Ising on a two-dimensional square lattice over $n$ vertices, that is, we deal with a discrete state space $\{$ 1, $\ldots,$ $Q\}^{n}$ for a positive integer $Q$ . Employing the Q-Ising (having a parameter $\beta$ ) as a prior distribution, and assuming a Gaussian noise (having another parameter a), a posterior is obtained from the Bayes’ formula. Furthermore, we generalize it: the distribution of noise is not necessarily a Gaussian, but any vertex-independent noise. We first present a Gibbs sampler from our posterior, and also present a perfect samplcr by defining a coupling via a monotone update function. Then, wc show $O(n\log n)$ mixing time of the Gibbs sampler for the generalized model under a condition that $\beta$ is sufficiently small (whatever the distribution of noise is). In case of a Gaussian, we obtain another more natural condition for rapid mixing that $\alpha$ is sufficiently larger than $\beta$ . Thereby, we show that the expected running time of our sampler is $O(n\log n)$ .