## A recursive algorithm for Markov random fields (2002)

Venue: | Biometrika |

Citations: | 12 - 1 self |

### BibTeX

@ARTICLE{Bartolucci02arecursive,

author = {Francesco Bartolucci and Julian Besag and Francesco Bartolucci and Julian Besag},

title = {A recursive algorithm for Markov random fields},

journal = {Biometrika},

year = {2002},

volume = {89},

pages = {724--730}

}

### OpenURL

### Abstract

The NRCSE was established in 1997 through a cooperative agreement with the United States Environmental Protection Agency which provides the Center's primary funding. A recursive algorithm for Markov random fields

### Citations

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Spatial interaction and the statistical analysis of lattice systems (with discussion
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(Show Context)
Citation Context ...tion Let X =(X1,...,Xn) denote a vector of n random variables, which here we take to be discrete. Let Si denote the minimal sample space for Xi and S that for X. Assume the positivity condition (e.g. =-=Besag, 1974-=-) that S = S1 × ...× Sn, which implies that any conditional probabilities we may wish to consider are well defined. A Markov random field for X is then a corresponding probability distribution {π(x) :... |

1132 | Graphical models
- Lauritzen
- 1996
(Show Context)
Citation Context ...ther Xj’s. A well–known example is the Ising model that occurs in statistical physics (Newman & Barkema, 1999, Ch. 1, 3, 4) but Markov random fields also play a central role in graphical models (e.g. =-=Lauritzen, 1996-=-), in random graphs (e.g. Frank and Strauss, 1986), in Markov chain Monte Carlo methods (e.g. Besag and Green, 1993; Smith and Roberts, 1993) and elsewhere. The requirements for a self–consistent spec... |

820 | A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains - Baum, Petrie, et al. - 1970 |

419 | Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms - Propp, Wilson - 1996 |

371 |
Bayesian computing via the Gibbs sampler and related Markov Chain Monte Carlo methods
- Smith, Roberts
- 1993
(Show Context)
Citation Context ...ov random fields also play a central role in graphical models (e.g. Lauritzen, 1996), in random graphs (e.g. Frank and Strauss, 1986), in Markov chain Monte Carlo methods (e.g. Besag and Green, 1993; =-=Smith and Roberts, 1993-=-) and elsewhere. The requirements for a self–consistent specification of π(.) via its full conditionals are not at all obvious but are identified by the Hammersley–Clifford theorem (Besag, 1974). Then... |

220 |
Markov graphs
- Frank, Strauss
- 1986
(Show Context)
Citation Context ...ing model that occurs in statistical physics (Newman & Barkema, 1999, Ch. 1, 3, 4) but Markov random fields also play a central role in graphical models (e.g. Lauritzen, 1996), in random graphs (e.g. =-=Frank and Strauss, 1986-=-), in Markov chain Monte Carlo methods (e.g. Besag and Green, 1993; Smith and Roberts, 1993) and elsewhere. The requirements for a self–consistent specification of π(.) via its full conditionals are n... |

162 | Correlation inequalities on some partially ordered sets - Fortuin, Kasteleyn, et al. - 1971 |

137 |
Bayesian computation and stochastic systems (with discussion
- Besag, Green, et al.
- 1995
(Show Context)
Citation Context ...andom fields. Larger systems, whether or not they are on a regular array, can be broken down into subsystems which are conditioned by their current boundary values, so that block Gibbs samplers (e.g. =-=Besag et al., 1995-=-, §2.4.5) can be devised. Finally, in §5, we show that, for multivariate binary distributions satisfying total positivity (Karlin and Rinott, 1980), our implementation of block Gibbs satisfies the mon... |

91 |
Monte Carlo methods in statistical physics
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- 1999
(Show Context)
Citation Context ...al statistics and there it is typical to choose the full conditional for each Xi to depend only on a few of the other Xj’s. A well–known example is the Ising model that occurs in statistical physics (=-=Newman & Barkema, 1999-=-, Ch. 1, 3, 4) but Markov random fields also play a central role in graphical models (e.g. Lauritzen, 1996), in random graphs (e.g. Frank and Strauss, 1986), in Markov chain Monte Carlo methods (e.g. ... |

65 |
Spatial Statistics and Bayesian Computation (with discussion
- Besag, Green
- 1993
(Show Context)
Citation Context ..., Ch. 1, 3, 4) but Markov random fields also play a central role in graphical models (e.g. Lauritzen, 1996), in random graphs (e.g. Frank and Strauss, 1986), in Markov chain Monte Carlo methods (e.g. =-=Besag and Green, 1993-=-; Smith and Roberts, 1993) and elsewhere. The requirements for a self–consistent specification of π(.) via its full conditionals are not at all obvious but are identified by the Hammersley–Clifford th... |

53 |
Classes of orderings of measures and related correlation inequalities i. multivariate totally positive distributions
- Karlin, Rinott
- 1980
(Show Context)
Citation Context ...urrent boundary values, so that block Gibbs samplers (e.g. Besag et al., 1995, §2.4.5) can be devised. Finally, in §5, we show that, for multivariate binary distributions satisfying total positivity (=-=Karlin and Rinott, 1980-=-), our implementation of block Gibbs satisfies the monotonicity condition of Propp and Wilson (1996), so that perfect block samplers can be constructed. For example, this applies to autologistic distr... |

53 | Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method - Robert, Rydén, et al. - 2000 |

17 | On the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbor systems - Brook - 1964 |

11 | Hidden Markov and Other Models for Discrete Valued Time Series - McDonald, Zucchini - 1997 |

8 | Some Approximations to the Binomial Distribution Function - Bahadur - 1960 |