### 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 language | English |
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Title of host publication | Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings |

Pages | 328-337 |

Number of pages | 10 |

DOIs | |

Publication status | Published - Dec 1 2009 |

Externally published | Yes |

Event | 15th Annual International Conference on Computing and Combinatorics, COCOON 2009 - Niagara Falls, NY, United States Duration: Jul 13 2009 → Jul 15 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5609 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th Annual International Conference on Computing and Combinatorics, COCOON 2009 |
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Country | United States |

City | Niagara Falls, NY |

Period | 7/13/09 → 7/15/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - A polynomial-time perfect sampler for the Q-ising with a vertex-independent noise

AU - Yamamoto, M.

AU - Kijima, S.

AU - Matsui, Y.

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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).

AB - 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).

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UR - http://www.scopus.com/inward/citedby.url?scp=76249108006&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-02882-3_33

DO - 10.1007/978-3-642-02882-3_33

M3 - Conference contribution

AN - SCOPUS:76249108006

SN - 3642028810

SN - 9783642028816

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 328

EP - 337

BT - Computing and Combinatorics - 15th Annual International Conference, COCOON 2009, Proceedings

ER -