A polynomial-time perfect sampler for the <Emphasis Type="Italic">Q</Emphasis>-Ising with a vertex-independent noise |
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Authors: | Masaki Yamamoto Shuji Kijima Yasuko Matsui |
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Institution: | 1.Dept. of Mathematical Sciences, School of Science,Tokai University,Tokyo,Japan;2.Research Institute for Mathematical Sciences,Kyoto University,Kyoto,Japan |
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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). |
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