Efficient recursions for general factorisable models.

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

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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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RAE_import_type : Journal article RAE_uoa_type : Statistics and Operational Research
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31 Mar 2008 08:57
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21 Nov 2022 19:18