Combining chains of Bayesian models with Markov melding

TL;DR

This paper introduces chained Markov melding, a method for combining Bayesian submodels linked in chains, effectively integrating heterogeneous data sources while addressing prior dependence and computational challenges.

Contribution

It extends Markov melding to chains of submodels, providing a new framework for Bayesian model integration with a specialized sampler for posterior estimation.

Findings

Successfully applied to ecological population models.

Effectively integrated longitudinal and time-to-event data.

Demonstrated computational efficiency in examples.

Abstract

A challenge for practitioners of Bayesian inference is specifying a model that incorporates multiple relevant, heterogeneous data sets. It may be easier to instead specify distinct submodels for each source of data, then join the submodels together. We consider chains of submodels, where submodels directly relate to their neighbours via common quantities which may be parameters or deterministic functions thereof. We propose chained Markov melding, an extension of Markov melding, a generic method to combine chains of submodels into a joint model. One challenge we address is appropriately capturing the prior dependence between common quantities within a submodel, whilst also reconciling differences in priors for the same common quantity between two adjacent submodels. Estimating the posterior of the resulting overall joint model is also challenging, so we describe a sampler that uses the chain structure to incorporate information contained in the submodels in multiple stages, possibly in parallel. We demonstrate our methodology using two examples. The first example considers an ecological integrated population model, where multiple data sets are required to accurately estimate population immigration and reproduction rates. We also consider a joint longitudinal and time-to-event model with uncertain, submodel-derived event times. Chained Markov melding is a conceptually appealing approach to integrating submodels in these settings.