Extensions of a Conflict Measure of Inconsistencies in Bayesian Hierarchical Models

Abstract

In a recent paper we extended and refined some tools introduced by O'Hagan for criticism of Bayesian hierarchical models. Especially, avoiding double use of data by a data-splitting approach was a main concern. Such tools can be applied at each node of the model, with a view to diagnosing problems of model fit at any point in the model structure. As O'Hagan, we investigated a Gaussian model of one-way analysis of variance. Through extensive Markov chain Monte Carlo simulations it was shown that our method detects model misspecification about as well as the one of O'Hagan, when this is properly calibrated, while retaining the desired false warning probability for data generated from the assumed model. In the present paper, we suggest some new measures of conflict based on tail probabilities of the so-called integrated posterior distributions introduced in our recent paper. These new measures are equivalent to the measure applied in the latter paper in simple Gaussian models, but seem more appropriately adjusted to deviations from normality and to conflicts not concerning location parameters. A general linear normal model with known covariance matrices is considered in detail. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.

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

“…These uniformity results indicate that the measures of Refs. [9] and [10] have comparable interpretations across different models. Therefore, they can be used without computationally costly precalibration schemes, such as the one suggested in Ref.…”

“…The general distribution of information sources given in Definition 2 is also introduced in Ref. [10]. For a given pair G p , G c of iic distributions, let λ Ã p and λ Ã c be independent samples from G p and G c , respectively.…”

“…For the conflict measures of Ref. [10], the uniform distribution on [0, 1] is the reference distribution. They focus on tail behavior but are based on the same iic distributions.…”

“…The nodes may be allowed to represent vectors, at both the parameter and the data levels [10]. Instead of DAGs, one may consider chain graphs, as described in Ref.…”

“…[10], these densities were instead used to define tail probability-based conflict measures. Gåsemyr and Natvig [10] showed that these measures are uniformly distributed in quite general hierarchical normal models with fixed variances/covariances. In Ref.…”