Node-Level Conflict Measures in Bayesian Hierarchical Models Based on Directed Acyclic Graphs

Abstract

Over the last decades, Bayesian hierarchical models defined by means of directed, acyclic graphs have become an essential and widely used methodology in the analysis of complex data. Simulation-based model criticism in such models can be based on conflict measures constructed by contrasting separate local information sources about each node in the graph. An initial suggestion of such a measure was not well calibrated. This shortcoming has, however, to a large extent been rectified by subsequently proposed alternative mutually similar tail probability-based measures, which have been proved to be uniformly distributed under the assumed model under various circumstances, and in particular, in quite general normal models with known covariance matrices. An advantage of this is that computationally costly precalibration schemes needed for some other suggested methods can be avoided. Another advantage is that noninformative prior distributions can be used when performing model criticism. In this chapter, we describe the basic framework and review the main uniformity results.

Alternatives to post‐processing posterior predictive p values

“…We will now establish the equivalence between the cppp / cepd approach and analysis based on conflict measures and , questioning the validity of the modeling assumption E ( μ ) = μ 0 , for model . Conflict concerning fixed top‐level nodes is introduced at the end of section 5 of Gåsemyr and Natvig () and summarized in section 4.4 of Gåsemyr and Natvig (). We choose to use this framework here, partly to highlight this as a useful variant of conflict analysis and partly because it allows for a derivation of exact analytic results.…”

“…In contrast, prior predictive testing can be avoided altogether if, instead, we can use the conflict measure approach introduced in Marshall and Spiegelhalter (2007), Dahl, Gåsemyr, and Natvig (2007), and Gåsemyr and Natvig (2009), nicely reviewed and applied in Presanis, Ohlssen, Spiegelhalter, and De Angelis (2013). Gåsemyr and Natvig (2017) provide a more recent review, also summarizing aspects of the theory appearing in Gåsemyr (2016). The basic idea behind a conflict measure is to consider potential conflict between the different sources of information regarding .…”

“…Applying this kind of reasoning more generally, a possibility in many cases might be to examine the degree of conflict between information about from data y versus prior information, without involving the discrepancy measure d( y, ) at all. The idea of assessing the degree of conflict between such information sources is pursued in O'Hagan (2003), which has been a source of inspiration for the measures introduced in Dahl et al (2007), Marshall and Spiegelhalter (2007), and Gåsemyr and Natvig (2009), reviewed and applied in Presanis et al (2013) and further analyzed in Gåsemyr (2016); see also the recent review by Gåsemyr and Natvig (2017). These measures can be applied for any parameter in a Bayesian model.…”

“…We use conflict about a top node in the model analyzed in Section 3.1, extending this sketchy described setup to a similar model with unknown precision. Both data node conflict and top node conflict are introduced and discussed in Gåsemyr and Natvig (2009) and summarized in Gåsemyr and Natvig (2017).…”