Combining Chains of Bayesian Models with Markov Melding

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.

Translating predictive distributions into informative priors

“…Setting a prior for models congruent with our knowledge is difficult without a method for translation such as we have proposed. We previously considered a survival model similar to the cure fraction model, where we knew a priori the fraction of cured/censored observations and a distribution of likely survival times, in our earlier work (Manderson and Goudie, 2022). Setting an appropriate prior for this model proved challenging, and would have benefited greatly from the translation methodology introduced in this paper.…”

Melding Wildlife Surveys to Improve Conservation Inference

“…The crux of fitting our integrated model was that the link parameters 𝒚 𝐴 and 𝒚 𝐺 are non-invertible functions of the submodel parameters 𝒛, 𝑵 𝐴 , and 𝑵 𝐺 . We adopted a chained Markov melding approach (Manderson & Goudie, 2022a) that facilitated joint inference for 𝒚 𝐴 and 𝒚 𝐺 accounting for the data, prior information, and assumptions in all three submodels. We derive the joint melded…”

“…Goudie et al (2019) recommended approximating the submodel marginals with kernel density estima-tors, but this approach can lead to numerical instabilities in implementation (Manderson & Goudie, 2022b). We obviated approximating the submodel marginal distributions by constructing [𝒚] pool using chained product of experts (PoE) pooling (Manderson & Goudie, 2022a),…”

“…Markov melding uses marginal replacement to form a melded posterior distribution for the link parameter that accounts for its implied prior and likelihood in each submodel. Recently, Manderson and Goudie (2022a) proposed chained Markov melding, an extension that facilitates joint inference for a sequence of submodels connected by multiple link parameters.…”

“…We facilitated shared inference of multiple LEPC surveys by chained Markov melding (Manderson & Goudie, 2022a) density estimates derived from submodels describing the observation processes of the aerial and ground surveys into a joint response model. Melding refines the submodel density estimates to those that agree with the spatio-temporal patterns observed in both surveys.…”