Recentered Importance Sampling With Applications to Bayesian Model Validation

Abstract

Since its introduction in the early 90's, the idea of using importance sampling (IS) with Markov chain Monte Carlo (MCMC) has found many applications. This paper examines problems associated with its application to repeated evaluation of related posterior distributions with a particular focus on Bayesian model validation. We demonstrate that, in certain applications, the curse of dimensionality can be reduced by a simple modification of IS. In addition to providing new theoretical insight into the behaviour of the IS approximation in a wide class of models, our result facilitates the implementation of computationally intensive Bayesian model checks. We illustrate the simplicity, computational savings and potential inferential advantages of the proposed approach through two substantive case studies, notably computation of Bayesian p-values for linear regression models and simulation-based model checking. Supplementary materials including appendices and the R code for Section 3.1.2 are available online.

Unbiased estimation of log normalizing constants with applications to Bayesian cross-validation

“…A number of articles discuss the computational difficulties associated with Bayesian cross-validation, e.g. Alqallaf and Gustafson (2001); Bhattacharya and Haslett (2007); Bornn et al (2010); Lamnisos et al (2012);McVinish et al (2013); Vehtari et al (2017). We first define the object of interest, before presenting our estimator.…”