Approximate Bayesian Inference with the Weighted Likelihood Bootstrap

SUMMARY We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution. This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares. In the generic weighting scheme, the WLB is first order correct under quite general conditions. Inaccuracies can be removed by using the WLB as a source of samples in the sampling-importance resampling (SIR) algorithm, which also allows incorporation of particular prior information. The SIR-adjusted WLB can be a competitive alternative to other integration methods in certain models. Asymptotic expansions elucidate the second-order properties of the WLB, which is a generalization of Rubin's Bayesian bootstrap. The calculation of approximate Bayes factors for model comparison is also considered. We note that, given a sample simulated from the posterior distribution, the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values; a modification of this estimator that avoids instability is also noted. These methods provide simple ways of calculating approximate Bayes factors and posterior model probabilities for a very wide class of models.