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Efficient recursions for general factorisable models

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  • R. Reeves

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(S-super-r+1) computations, as opposed to O(S-super-n) 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. Copyright Biometrika Trust 2004, Oxford University Press.

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  • R. Reeves, 2004. "Efficient recursions for general factorisable models," Biometrika, Biometrika Trust, vol. 91(3), pages 751-757, September.
    • Handle: RePEc:oup:biomet:v:91:y:2004:i:3:p:751-757
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    Cited by:

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    3. Cécile Hardouin & Xavier Guyon, 2014. "Recursions on the marginals and exact computation of the normalizing constant for Gibbs processes," Computational Statistics, Springer, vol. 29(6), pages 1637-1650, December.
    4. Solaiman Afroughi & Soghrat Faghihzadeh & Majid Jafari Khaledi & Mehdi Ghandehari Motlagh & Ebrahim Hajizadeh, 2011. "Analysis of clustered spatially correlated binary data using autologistic model and Bayesian method with an application to dental caries of 3--5-year-old children," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(12), pages 2763-2774, February.
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    6. Magnussen, Steen & Reeves, Rob, 2008. "A method for bias-reduction of sample-based MLE of the autologistic model," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 103-111, September.
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    9. Nial Friel & Håvard Rue, 2007. "Recursive computing and simulation-free inference for general factorizable models," Biometrika, Biometrika Trust, vol. 94(3), pages 661-672.

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