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Computation of marginal likelihoods with data-dependent support for latent variables. (English) Zbl 1471.62089

Summary: Several Monte Carlo methods have been proposed for computing marginal likelihoods in Bayesian analyses. Some of these involve sampling from a sequence of intermediate distributions between the prior and posterior. A difficulty arises if the support in the posterior distribution is a proper subset of that in the prior distribution. This can happen in problems involving latent variables whose support depends upon the data and can make some methods inefficient and others invalid. The correction required for models of this type is derived and its use is illustrated by finding the marginal likelihoods in two examples. One concerns a model for competing risks. The other involves a zero-inflated over-dispersed Poisson model for counts of centipedes, using latent Gaussian variables to capture spatial dependence.

MSC:

62-08 Computational methods for problems pertaining to statistics
62F15 Bayesian inference
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[1] Blackburn, J.; Farrow, M.; Arthur, W., Factors influencing the distribution, abundance and diversity of geophilomorph and lithobiomorph centipedes, J. Zool., 256, 221-232, (2002)
[2] Chib, S., Marginal likelihood from the Gibbs output, J. Amer. Statist. Assoc., 90, 1313-1321, (1995) · Zbl 0868.62027
[3] Chib, S.; Greenberg, E., Analysis of multivariate probit models, Biometrika, 85, 347-361, (1998) · Zbl 0938.62020
[4] Chib, S.; Jeliazkov, I., Marginal likelihood from the metropolis-Hastings output, J. Amer. Statist. Assoc., 96, 270-281, (2001) · Zbl 1015.62020
[5] Crowder, M. J., Classical competing risks, (2001), Chapman and Hall/CRC London · Zbl 0979.62078
[6] Diggle, P. J.; Ribeiro, P. J., (Model-Based Geostatistics, Springer Series in Statistics, (2007), Springer-Verlag New York) · Zbl 1132.86002
[7] Friel, N., Hurn, M., Wyse, J., 2012. Improving power posterior estimation of statistical evidence. arXiv Pre-print arXiv:1209.3198 [stat.CO].
[8] Friel, N.; Pettitt, A. N., Marginal likelihood estimation via power posteriors, J. R. Stat. Soc. Ser. B, 70, 589-607, (2008) · Zbl 05563360
[9] Friel, N.; Wyse, J., Estimating the evidence—a review, Stat. Neerl., 66, 288-308, (2012)
[10] Garthwaite, P. H.; Kadane, J. B.; O’Hagan, A., Statistical methods for eliciting probability distributions, J. Amer. Statist. Assoc., 100, 680-700, (2005) · Zbl 1117.62340
[11] Gelman, A.; Meng, X. L., Simulating normalizing constants: from importance sampling to bridge sampling to path sampling, Statist. Sci., 13, 163-185, (1998) · Zbl 0966.65004
[12] Genz, A., Numerical computation of multivariate normal probabilities, J. Comput. Graph. Statist., 1, 141-149, (1992)
[13] Genz, A.; Bretz, F., (Computation of Multivariate Normal and t Probabilities, Lecture Notes in Statistics, vol. 195, (2009), Springer-Verlag Heidelberg) · Zbl 1204.62088
[14] Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., Hothorn, T., 2012. mvtnorm: multivariate normal and t distributions. R package version 0.9-9992.
[15] Germain, S.E., 2010. Bayesian spatio-temporal modelling of rainfall through non-homogeneous hidden Markov models. Ph.D. Thesis. School of Mathematics & Statistics, Newcastle University, UK.
[16] Hoel, D. G.; Walburg, H. E., Statistical analysis of survival experiments, J. Natl. Cancer Inst., 49, 361-372, (1972)
[17] Johnson, S.G., Narasimhan, B., 2011. cubature: adaptive multivariate integration over hypercubes. R package version 1.1-1.
[18] McCulloch, R. E.; Polson, N. G.; Rossi, P. E., A Bayesian analysis of the multinomial probit model with fully identified parameters, J. Econometrics, 99, 173-193, (2000) · Zbl 0958.62029
[19] Meng, X. L.; Wong, W. H., Simulating ratios of normalizing constants via a simple identity: a theoretical exploration, Statist. Sinica, 6, 831-860, (1996) · Zbl 0857.62017
[20] Neal, R. M., Annealed importance sampling, Stat. Comput., 11, 125-139, (2001)
[21] Neal, R. M., Estimating ratios of normalizing constants using linked importance sampling. technical report 0511. department of statistics, university of Toronto, (2005)
[22] (R: A Language and Environment for Statistical Computing, (2012), R Foundation for Statistical Computing Vienna, Austria), R Development Core Team
[23] Schmidt, A. M.; Guttorp, P.; O’Hagan, A., Considering covariates in the covariance structure of spatial processes, Environmetrics, 22, 487-500, (2011)
[24] Schmidt, A. M.; Rodríguez, M. A., Modelling multivariate counts varying continuously in space (with discussion), (Bernardo, J. M.; Bayarri, M. J.; Berger, J. O.; Dawid, A. P.; Heckerman, D.; Smith, A. F.M.; West, M., Bayesian Statistics, Vol. 9, (2011), Oxford University Press), 611-638
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