Open Access
September 2023 Combining Chains of Bayesian Models with Markov Melding
Andrew A. Manderson, Robert J. B. Goudie
Author Affiliations +
Bayesian Anal. 18(3): 807-840 (September 2023). DOI: 10.1214/22-BA1327


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.

Funding Statement

This work was supported by The Alan Turing Institute under the UK Engineering and Physical Sciences Research Council (EPSRC) [EP/N510129/1] and the UK Medical Research Council [programme code MC_UU_00002/2].


We thank Sarah L Cowan for assistance in understanding respiratory failure, and Anne Presanis and Brian Tom for many helpful discussions about the methodological aspects of this work. We also thank Luiz Max Carvalho for comments on an earlier version of this paper.


Download Citation

Andrew A. Manderson. Robert J. B. Goudie. "Combining Chains of Bayesian Models with Markov Melding." Bayesian Anal. 18 (3) 807 - 840, September 2023.


Published: September 2023
First available in Project Euclid: 24 August 2022

MathSciNet: MR4626358
Digital Object Identifier: 10.1214/22-BA1327

Keywords: Bayesian graphical models , combining models , integrated population model , Markov melding , model/data integration , multi-stage estimation

Vol.18 • No. 3 • September 2023
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