Computational Statistics and Data Analysis 52 (2008) 4608–4624 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Marginal likelihoods for non-Gaussian models using auxiliary mixture sampling Sylvia Frühwirth-Schnatter ∗ , Helga Wagner Johannes Kepler Universität Linz, Department of Applied Statistics and Econometrics, Altenbergerstraße 69, A-4040 Linz, Austria article info Article history: Received 13 June 2007 Received in revised form 13 March 2008 Accepted 14 March 2008 Available online 3 April 2008 abstract Several new estimators of the marginal likelihood for complex non-Gaussian models are developed. These estimators make use of the output of auxiliary mixture sampling for count data and for binary and multinomial data. One of these estimators is based on combining Chib’s estimator with data augmentation as in auxiliary mixture sampling, while the other estimators are importance sampling and bridge sampling based on constructing an unsupervised importance density from the output of auxiliary mixture sampling. These estimators are applied to a logit regression model, to a Poisson regression model, to a binomial model with random intercept, as well as to state space modeling of count data. © 2008 Elsevier B.V. All rights reserved. 1. Introduction For an applied statistician, choosing an appropriate model from a class of candidate models is a fundamental data analytical task. In a classical framework, model selection problems are addressed either via hypothesis testing for nested models, or by using information criteria such as the Akaike information criterion (Akaike, 1974) or Schwarz’s criterion or BIC (Schwarz, 1978). In a Bayesian setting, model selection relies on the posterior probabilities of a model given the data, see Bernardo and Smith (1994) as well as the recent reviews by Godsill (2001), Green (2003) and Kadane and Lazar (2004). More formally, suppose there are K different models M 1 ,..., M K , which are candidates for having generated the data y. Each of these models is assigned a prior probability p(M k ) and the goal is to derive the posterior model probabilities p(M k |y) for each model M k , k = 1,..., K. There are basically two strategies to implement Bayesian model selection. Model space MCMC methods directly sample from the discrete model space (M 1 ,..., M K ) by drawing jointly model indicators and parameters, using e.g. the reversible jump MCMC algorithm (Green, 1995) or the stochastic variable selection approach (George and McCulloch, 1993, 1997). A more classical strategy which dates back to Jeffreys (1948) and Zellner (1971) determines the posterior model probabilities p(M k |y) separately for each model by using Bayes’ rule: p(M k |y) ∝ p(y|M k )p(M k ), where p(y|M k ) is the marginal likelihood of model M k . Let ϑ k denote the vector containing the unknown parameters of model M k . Then the marginal likelihood is given as p(y|M k ) =  Θ k p(y|ϑ k )p(ϑ k )dϑ k , (1) with p(ϑ k ) being the prior distribution of model parameter ϑ k with support Θ k . ∗ Corresponding author. Tel.: +43 732 2468 8294; fax: +43 732 2468 9846. E-mail address: sylvia.fruehwirth-schnatter@jku.at (S. Frühwirth-Schnatter). 0167-9473/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2008.03.028