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Abstract
Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first-order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.
Highlights
Transdimensional Markov chain Monte Carlo (MCMC) methods provide an indispensable tool for the Bayesian analysis of models with varying dimensionality (Sisson 2005)
In order to ensure that the Markov chain converges to the correct stationary distribution, transdimensional MCMC methods such as reversible jump MCMC (Green 1995) or the product space approach (Carlin and Chib 1995) match the dimensionality of parameter spaces across different models
Transdimensional MCMC methods have proven to be very useful for the analysis of many statistical models including capture–recapture models (Arnold et al 2010), generalized linear models (Forster et al 2012), factor models (Lopes and West 2004), and mixture models (Frühwirth-Schnatter 2001), and are widely used in substantive applications such as selection of phylogenetic trees (Opgen-Rhein et al 2005), gravitational wave detection in Statistics and Computing (2019) 29:631–643 (a) Independent Sampling (b) Markov chain
Summary
Transdimensional Markov chain Monte Carlo (MCMC) methods provide an indispensable tool for the Bayesian analysis of models with varying dimensionality (Sisson 2005). T , posterior samples are obtained for the indexing variable z(t) and the model parameters, which are usually continuous and differ in dimensionality (for a review, see Sisson 2005). We propose to fit a discrete, first-order Markov model to the MCMC output z(t) to assess the precision of the estimated stationary distribution π. Whereas several diagnostics have previously been proposed to assess the convergence of transdimensional MCMC samplers (e.g., Brooks and Giudici 2000; Castelloe and Zimmerman 2002; Brooks et al 2003a; Sisson and Fan 2007), we are unaware of any methods that quantify the precision of the point estimate π
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