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(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).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bubley, R., Dyer, M.: Path coupling: A technique for proving rapid mixing in Markov chains. In: Proc. of FOCS 1997, pp. 223–231 (1997)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions,and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence 6, 721–741 (1984)
Gibbs, A.L.: Bounding the convergence time of the Gibbs sampler in Bayesian image restoration. Biometrika 87(4), 749–766 (2000)
Gibbs, A.L.: Convergence in the Wasserstein metric for Markov chain Monte Carlo algorithms with applications to image restoration. Stochastic Models 20, 473–492 (2004)
Inoue, J., Carlucci, D.M.: Image restoration using the Q-Ising spin glass. Phys. Rev. E 64, 036121 (2001)
Propp, J., Wilson, D.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. and Algo. 9, 223–252 (1996)
Tanaka, K., Inoue, J., Titterington, D.M.: Probabilistic image processing by means of the Bethe approximation for the Q-Ising model. J. Phys. A: Math. Gen. 36, 11023–11035 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yamamoto, M., Kijima, S., Matsui, Y. (2009). A Polynomial-Time Perfect Sampler for the Q-Ising with a Vertex-Independent Noise. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-02882-3_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02881-6
Online ISBN: 978-3-642-02882-3
eBook Packages: Computer ScienceComputer Science (R0)