Computing bayes factors from data with missing values

The Bayes factor is increasingly used for the evaluation of hypotheses. These may be traditional hypotheses specified using equality constraints among the parameters of the statistical model of interest or informative hypotheses specified using equality and inequality constraints. Thus far, no attention has been given to the computation of Bayes factors from data with missing values. A key property of such a Bayes factor should be that it is only based on the information in the observed values. This article will show that such a Bayes factor can be obtained using multiple imputations of the missing values. After introduction of the general framework elaborations for Bayes factors based on default or subjective prior distributions and Bayes factors based on priors specified using training data will be given. It will be illustrated that the approach proposed can be applied using R packages for multiple imputation in combination with the Bayes factor packages Bain and BayesFactor. It will furthermore be illustrated that Bayes factors computed using a single imputation of the data are very inaccurate approximations of the correct Bayes factor.