Pettitt, Anthony and Reeves, R. (2004) *Efficient recursions for general factorisable models.* Biometrika, 91 (3). pp. 751-757. ISSN 1464-3510

## 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.

Item Type: | Article |
---|---|

Journal or Publication Title: | Biometrika |

Additional Information: | RAE_import_type : Journal article RAE_uoa_type : Statistics and Operational Research |

Uncontrolled Keywords: | Autologistic distribution ; Gibbs distribution ; Ising model ; Normalising constant ; Partition function ; Markov chain Monte Carlo. |

Subjects: | ?? qa ?? |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 2434 |

Deposited By: | ep_importer |

Deposited On: | 31 Mar 2008 09:57 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 22 Jul 2017 01:31 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/2434 |

### Actions (login required)

View Item |