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Marginal likelihood estimation via power posteriors

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  • N. Friel
  • A. N. Pettitt

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.

Suggested Citation

  • N. Friel & A. N. Pettitt, 2008. "Marginal likelihood estimation via power posteriors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 589-607.
  • Handle: RePEc:bla:jorssb:v:70:y:2008:i:3:p:589-607
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    References listed on IDEAS

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    1. 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.
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    7. 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.
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    Cited by:

    1. repec:bla:jorssb:v:79:y:2017:i:3:p:695-718 is not listed on IDEAS
    2. 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.
    3. 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.
    4. Chan, Joshua C.C. & Grant, Angelia L., 2016. "Fast computation of the deviance information criterion for latent variable models," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 847-859.
    5. Joshua C. C. Chan & Eric Eisenstat, 2015. "Marginal Likelihood Estimation with the Cross-Entropy Method," Econometric Reviews, Taylor & Francis Journals, vol. 34(3), pages 256-285, March.
    6. Xing Ju Lee & Christopher C. Drovandi & Anthony N. Pettitt, 2015. "Model choice problems using approximate Bayesian computation with applications to pathogen transmission data sets," Biometrics, The International Biometric Society, vol. 71(1), pages 198-207, March.
    7. 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.
    8. 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.
    9. repec:dau:papers:123456789/5724 is not listed on IDEAS
    10. Loza-Reyes, E. & Hurn, M.A. & Robinson, A., 2014. "Classification of molecular sequence data using Bayesian phylogenetic mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 81-95.
    11. Christopher C. Drovandi & Anthony N. Pettitt, 2013. "Bayesian Experimental Design for Models with Intractable Likelihoods," Biometrics, The International Biometric Society, vol. 69(4), pages 937-948, December.
    12. 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.
    13. Jeong Eun Lee & Christian Robert, 2013. "Imortance Sampling Schemes for Evidence Approximation in Mixture Models," Working Papers 2013-42, Center for Research in Economics and Statistics.
    14. Perrakis, Konstantinos & Ntzoufras, Ioannis & Tsionas, Efthymios G., 2014. "On the use of marginal posteriors in marginal likelihood estimation via importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 54-69.
    15. repec:bla:jorssb:v:79:y:2017:i:2:p:323-380 is not listed on IDEAS
    16. repec:eee:csdana:v:122:y:2018:i:c:p:33-44 is not listed on IDEAS
    17. 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.
    18. 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.
    19. repec:eee:phsmap:v:486:y:2017:i:c:p:587-594 is not listed on IDEAS
    20. repec:eee:reensy:v:176:y:2018:i:c:p:52-61 is not listed on IDEAS
    21. Vitoratou, Silia & Ntzoufras, Ioannis & Moustaki, Irini, 2016. "Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions," LSE Research Online Documents on Economics 57685, London School of Economics and Political Science, LSE Library.
    22. Ehlers, Ricardo S., 2012. "Computational tools for comparing asymmetric GARCH models via Bayes factors," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(5), pages 858-867.
    23. repec:bla:jorssc:v:66:y:2017:i:4:p:869-890 is not listed on IDEAS

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