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Marginal likelihood estimation via power posteriors

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  • N. Friel
  • A. N. Pettitt
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    Abstract

    Model choice plays an increasingly important role in statistics. From a Bayesian perspective a crucial goal is to compute the marginal likelihood of the data for a given model. However, this is typically a difficult task since it amounts to integrating over all model parameters. The aim of the paper is to illustrate how this may be achieved by using ideas from thermodynamic integration or path sampling. We show how the marginal likelihood can be computed via Markov chain Monte Carlo methods on modified posterior distributions for each model. This then allows Bayes factors or posterior model probabilities to be calculated. We show that this approach requires very little tuning and is straightforward to implement. The new method is illustrated in a variety of challenging statistical settings. Copyright (c) 2008 Royal Statistical Society.

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    Bibliographic Info

    Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

    Volume (Year): 70 (2008)
    Issue (Month): 3 ()
    Pages: 589-607

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    Handle: RePEc:bla:jorssb:v:70:y:2008:i:3:p:589-607

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    References

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    1. Ian L. Dryden & Mark R. Scarr & Charles C. Taylor, 2003. "Bayesian texture segmentation of weed and crop images using reversible jump Markov chain Monte Carlo methods," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 52(1), pages 31-50.
    2. Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
    3. Sisson, Scott A., 2005. "Transdimensional Markov Chains: A Decade of Progress and Future Perspectives," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1077-1089, September.
    4. R. Reeves, 2004. "Efficient recursions for general factorisable models," Biometrika, Biometrika Trust, vol. 91(3), pages 751-757, September.
    5. Francesco Bartolucci & Luisa Scaccia & Antonietta Mira, 2006. "Efficient Bayes factor estimation from the reversible jump output," Biometrika, Biometrika Trust, vol. 93(1), pages 41-52, March.
    6. Jose M. Perez, 2002. "Expected-posterior prior distributions for model selection," Biometrika, Biometrika Trust, vol. 89(3), pages 491-512, August.
    7. S. P. Brooks & P. Giudici & G. O. Roberts, 2003. "Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 3-39.
    8. P. G. Ridall & A. N. Pettitt & N. Friel & P. A. McCombe & R. D. Henderson, 2007. "Motor unit number estimation using reversible jump Markov chain Monte Carlo methods," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 56(3), pages 235-269.
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    Cited by:
    1. Drovandi, Christopher C. & McGree, James M. & Pettitt, Anthony N., 2013. "Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 320-335.
    2. Heaps, Sarah E. & Boys, Richard J. & Farrow, Malcolm, 2014. "Computation of marginal likelihoods with data-dependent support for latent variables," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 392-401.
    3. Spezia, L. & Cooksley, S.L. & Brewer, M.J. & Donnelly, D. & Tree, A., 2014. "Modelling species abundance in a river by Negative Binomial hidden Markov models," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 599-614.
    4. Calderhead, Ben & Girolami, Mark, 2009. "Estimating Bayes factors via thermodynamic integration and population MCMC," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4028-4045, October.
    5. Ryan, Elizabeth G. & Drovandi, Christopher C. & Thompson, M. Helen & Pettitt, Anthony N., 2014. "Towards Bayesian experimental design for nonlinear models that require a large number of sampling times," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 45-60.
    6. Chan, Joshua & Eisenstat, Eric, 2012. "Marginal Likelihood Estimation with the Cross-Entropy Method," MPRA Paper 40051, University Library of Munich, Germany.
    7. Lefebvre, Geneviève & Steele, Russell & Vandal, Alain C., 2010. "A path sampling identity for computing the Kullback-Leibler and J divergences," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1719-1731, July.
    8. Jeong Eun Lee & Christian Robert, 2013. "Imortance Sampling Schemes for Evidence Approximation in Mixture Models," Working Papers 2013-42, Centre de Recherche en Economie et Statistique.
    9. Filippone, Maurizio & Sanguinetti, Guido, 2011. "Approximate inference of the bandwidth in multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3104-3122, December.
    10. Joshua C.C. Chan & Angelia L. Grant, 2014. "Fast Computation of the Deviance Information Criterion for Latent Variable Models," CAMA Working Papers 2014-09, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.

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