A numerically stable algorithm for integrating Bayesian models using Markov melding
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When statistical analyses consider multiple data sources, Markov melding provides a method for combining the source-specific Bayesian models. Models often contain different quantities of information due to variation in the richness of model-specific data, or availability of model-specific prior information. We show that this can make the multi-stage Markov chain Monte Carlo sampler employed by Markov melding unstable and unreliable. We propose a robust multi-stage algorithm that estimates the required prior marginal self-density ratios using weighted samples, dramatically improving accuracy in the tails of the distribution, thus stabilising the algorithm and providing reliable inference. We demonstrate our approach using an evidence synthesis for inferring HIV prevalence, and an evidence synthesis of A/H1N1 influenza.
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