# Efficient recursions for general factorisable models

Reeves, Robert W. & Pettitt, Anthony N.
(2004)
Efficient recursions for general factorisable models.
*Biometrika*, *91*(3), pp. 751-757.

## Abstract

Let n S-valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant. For an unnormalised joint likelihood expressible as a product of factors, we give an algebraic recursion which can be used for computing the normalising constant and other summations. A saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution, with maximum computational efficiency as the subsets attain their minimum size. If each subset contains at most r+1 of the n components in the joint distribution, we term this a lag-r model, whose normalising constant can be computed using a forward recursion in O(Sr+1) computations, as opposed to O(Sn) for the direct computation. We show how a lag-r model represents a Markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood. We illustrate the method by showing how the normalising constant of the Ising or autologistic model can be computed.

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ID Code: | 10211 |
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Item Type: | Journal Article |

Refereed: | Yes |

DOI: | 10.1093/biomet/91.3.751 |

ISSN: | 0006-3444 |

Divisions: | Past > QUT Faculties & Divisions > Faculty of Science and Technology |

Copyright Owner: | Copyright 2004 Oxford University Press |

Copyright Statement: | This is an electronic version of an article published in Reeves, Robert W. and Pettitt, Tony N. (2004) Efficient recursions for general factorisable models . Biometrika 91(3):pp. 751-757. |

Deposited On: | 17 Oct 2007 |

Last Modified: | 29 Feb 2012 13:09 |

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