Prior-Proposal Recursive Bayesian Inference
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Bayesian models are naturally equipped to provide recursive inference 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. Recursive Bayesian methods include two main approaches that we refer to as Prior- and Proposal-Recursive Bayes. Prior-Recursive Bayes uses Bayesian updating, fitting models to partitions of data sequentially, and provides a convenient way to accommodate new data as they become available. Prior-Recursive Bayes uses the posterior from the previous stage as the prior in the new stage based on the latest data. By contrast, 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 posterior distributions from the first stage as proposals in an MCMC algorithm to fit the full model. The second-stage recursive proposals simplify the Metropolis-Hastings ratio substantially and can lead to computational advantages for the Proposal-Recursive Bayes method. We combine Prior- and Proposal-Recursive concepts in a framework that can be used to fit any Bayesian model exactly, and often with computational improvements. We demonstrate our new method by fitting a geostatistical model to spatially-explicit data in a sequence of stages, leading to computational improvements by a factor of three in our example. While the method we propose provides exact inference, it can also be coupled with modern approximation methods leading to additional computational efficiency. Overall, our new approach has implications for big data, streaming data, and optimal adaptive design situations and can be modified to fit a broad class of Bayesian models to data.
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