A Note on Bayesian Inference After Multiple Imputation

This article is aimed at practitioners who plan to use Bayesian inference on multiply-imputed datasets in settings where posterior distributions of the parameters of interest are not approximately Gaussian. We seek to steer practitioners away from a naive approach to Bayesian inference, namely estimating the posterior distribution in each completed dataset and averaging functionals of these distributions. We demonstrate that this approach results in unreliable inferences. A better approach is to mix draws from the posterior distributions from each completed dataset, and use the mixed draws to summarize the posterior distribution. Using simulations, we show that for this second approach to work well, the number of imputed datasets should be large. In particular, five to ten imputed datasets which is the standard recommendation for multiple imputation is generally not enough to result in reliable Bayesian inferences.