Multiple imputation is a common approach for handling missing data. It is also used by government agencies to protect confidential information in public use data files. One reason for the popularity of multiple imputation approaches is ease of use: Analysts make inferences by combining point and variance estimates with simple rules. These combining rules are based on method of moments approximations to full Bayesian inference. With modern computing, however, it is as easy to perform the full Bayesian inference as it is to combine point and variance estimates. This begs the question: Is there any advantage of using full Bayesian inference over multiple imputation combining rules? We use simulation studies to investigate this question. We find that, in general, the full Bayesian inference is not preferable to using the combining rules in multiple imputation for missing data. The full Bayesian inference can have advantages over the combining rules when using multiple imputation to protect confidential information.