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

M. Yamamoto, S. Kijima, Y. Matsui

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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).

Original languageEnglish
Title of host publicationComputing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings
Pages328-337
Number of pages10
DOIs
Publication statusPublished - Dec 1 2009
Externally publishedYes
Event15th Annual International Conference on Computing and Combinatorics, COCOON 2009 - Niagara Falls, NY, United States
Duration: Jul 13 2009Jul 15 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5609 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other15th Annual International Conference on Computing and Combinatorics, COCOON 2009
CountryUnited States
CityNiagara Falls, NY
Period7/13/097/15/09

Fingerprint

Ising
Polynomial time
Polynomials
Statistical mechanics
Vertex of a graph
Image reconstruction
Gibbs Sampler
Q-integers
Bayes' Formula
Mixing Time
Image Restoration
Gaussian Noise
Prior distribution
Square Lattice
Statistical Mechanics
Monotone
State Space
Update
Generalise
Model

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Yamamoto, M., Kijima, S., & Matsui, Y. (2009). A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise. In Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings (pp. 328-337). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5609 LNCS). https://doi.org/10.1007/978-3-642-02882-3_33

A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise. / Yamamoto, M.; Kijima, S.; Matsui, Y.

Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings. 2009. p. 328-337 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5609 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Yamamoto, M, Kijima, S & Matsui, Y 2009, A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise. in Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5609 LNCS, pp. 328-337, 15th Annual International Conference on Computing and Combinatorics, COCOON 2009, Niagara Falls, NY, United States, 7/13/09. https://doi.org/10.1007/978-3-642-02882-3_33
Yamamoto M, Kijima S, Matsui Y. A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise. In Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings. 2009. p. 328-337. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-02882-3_33
Yamamoto, M. ; Kijima, S. ; Matsui, Y. / A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise. Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings. 2009. pp. 328-337 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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