A polynomial-time perfect sampler for the Q-Ising with a vertex-independent noise

Masaki Yamamoto (), Shuji Kijima () and Yasuko Matsui ()
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Masaki Yamamoto: Tokai University
Shuji Kijima: Kyoto University
Yasuko Matsui: Tokai University

Journal of Combinatorial Optimization, 2011, vol. 22, issue 3, No 7, 392-408

Abstract: Abstract 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(nlog n) 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(nlog n).

Keywords: Markov chain Monte Carlo; Perfect sampler; Q-Ising; Vertex-independent noise (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10878-010-9309-7

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