Abstract
We propose an easily computed estimator of the marginal likelihood from posterior simulation output, via reciprocal importance sampling, combining earlier proposals of DiCiccio et al (1997) and Robert and Wraith (2009). This involves only the unnormalized posterior densities from the sampled parameter values, and does not involve additional simulations beyond the main posterior simulation, or additional complicated calculations, provided that the parameter space is unconstrained. Even if this is not the case, the estimator is easily adjusted by a simple Monte Carlo approximation. It is unbiased for the reciprocal of the marginal likelihood, consistent, has finite variance, and is asymptotically normal. It involves one user-specified control parameter, and we derive an optimal way of specifying this. We illustrate it with several numerical examples.
Funding Statement
Irons’s research was supported by a Shanahan Endowment Fellowship and a Eunice Kennedy Shriver National Institute of Child Health and Human Development training grant, T32 HD101442-01, to the Center for Studies in Demography & Ecology at the University of Washington. Raftery’s research was supported by NIH grant R01 HD070936 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), by the Fondation des Sciences Mathématiques de Paris (FSMP), and by Université Paris-Cité.
Acknowledgments
The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments. The authors would further like to thank Christoph Richard from the Friedrich-Alexander-Universität Erlangen-Nürnberg for his helpful comments. Their comments dramatically improved the content and quality of this paper.
Citation
Martin Metodiev. Marie Perrot-Dockès. Sarah Ouadah. Nicholas J. Irons. Pierre Latouche. Adrian E. Raftery. "Easily Computed Marginal Likelihoods from Posterior Simulation Using the THAMES Estimator." Bayesian Anal. Advance Publication 1 - 28, 2024. https://doi.org/10.1214/24-BA1422
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