Combining datasets for Bayesian inference

Bayesian inference (BI) has long been used to estimate posterior parameter distributions based on multiple sets of data. Here, we present elementary derivations of two strategies for doing so. The first approach employs the posterior distribution from a BI calculation on one dataset as the prior distribution for a second dataset. The second approach uses importance sampling, augmenting the posterior from one dataset to form a well-targeted sampling function for the second. In both cases, the distribution sampled is shown to be the ``full posterior'' as if BI were performed on the two datasets together, subject to only mild assumptions. Both methods can be applied in sequence to multiple datasets.