  Bayesian models provide recursive inference naturally because they can
formally reconcile new data and existing scientific information. However,
popular use of Bayesian methods often avoids priors that are based on exact
posterior distributions resulting from former studies. Two existing Recursive
Bayesian methods are: Prior- and Proposal-Recursive Bayes. Prior-Recursive
Bayes uses Bayesian updating, fitting models to partitions of data
sequentially, and provides a way to accommodate new data as they become
available using the posterior from the previous stage as the prior in the new
stage based on the latest data. Proposal-Recursive Bayes is intended for use
with hierarchical Bayesian models and uses a set of transient priors in first
stage independent analyses of the data partitions. The second stage of
Proposal-Recursive Bayes uses the posteriors from the first stage as proposals
in an MCMC algorithm to fit the full model. We combine Prior- and
Proposal-Recursive concepts to fit any Bayesian model, and often with
computational improvements. We demonstrate our method with two case studies.
Our approach has implications for big data, streaming data, and optimal
adaptive design situations.
