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

Masaki Yamamoto, Shuji Kijima, Yasuko Matsui

Research output: Contribution to journalArticle

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).

Original languageEnglish
Pages (from-to)392-408
Number of pages17
JournalJournal of Combinatorial Optimization
Volume22
Issue number3
DOIs
Publication statusPublished - Oct 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics

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