## 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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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) :... |

1102 |
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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... |

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Bayesian computation via the Gibbs Sampler and related Markov Chain Monte Carlo Method
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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... |

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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... |

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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... |

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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. ... |

61 |
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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... |

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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... |

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