Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis

Abstract

Complex stochastic models represented by directed acyclic graphs (DAGs) are increasingly employed to synthesise multiple, imperfect and disparate sources of evidence, to estimate quantities that are difficult to measure directly. The various data sources are dependent on shared parameters and hence have the potential to conflict with each other, as well as with the model. In a Bayesian framework, the model consists of three components: the prior distribution, the assumed form of the likelihood and structural assumptions. Any of these components may be incompatible with the observed data. The detection and quantification of such conflict and of data sources that are inconsistent with each other is therefore a crucial component of the model criticism process. We first review Bayesian model criticism, with a focus on conflict detection, before describing a general diagnostic for detecting and quantifying conflict between the evidence in different partitions of a DAG. The diagnostic is a $p$-value based on splitting the information contributing to inference about a “separator” node or group of nodes into two independent groups and testing whether the two groups result in the same inference about the separator node(s). We illustrate the method with three comprehensive examples: an evidence synthesis to estimate HIV prevalence; an evidence synthesis to estimate influenza case-severity; and a hierarchical growth model for rat weights.