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,…,Q}
n
for a positive integer Q. Employing the Q‐Ising (having a parameter β) as a prior distribution, and assuming a Gaussian noise (having another parameter α), 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 sampler by defining a coupling via a monotone update function. Then, we show O(nlogn) mixing time of the Gibbs sampler for the generalized model under a condition that β is sufficiently small (whatever the distribution of noise is). In case of a Gaussian, we obtain another more natural condition for rapid mixing that α is sufficiently larger than β. Thereby, we show that the expected running time of our sampler is O(nlogn).
