Graphical and Summary Diagnostics for Node Level Adequacy in Bayesian Hierarchical Models

For Bayesian hierarchical models represented by directed acyclic graphs, all neighbours of a node provide, potentially conflicting, information on the node. Hence identifying potential conflict between these different information sources can be important for assessing the adequacy of Bayesian hierarchical models. Building on this idea originating from O’Hagan (2003), Gåsemyr and Natvig (2009) constructed node based conflict measures that have been shown to be well calibrated. Sharing the idea of considering potential conflict between separate information sources for each node, Scheel et al. (2011) constructed the graphical diagnostic the local critique plot (lcp). This can be used both to identify conflict at each node, and provide insight into the nature of the conflict, hence being more informative than summary diagnostics. In the present paper we link these two ideas together by constructing a new diagnostic plot, the integrated information contributions local critique plot (iic-lcp), that is supplementary to lcp, but builds on the framework of Gåsemyr and Natvig (2009). It has the advantage over the lcp with the possibility to display curves corresponding to different parameters in the same plot, saving space and easing comparisons, particularly useful for sets of parameters representing exchangeable groups. Notably, we establish a relationship between the geometry of the iic-lcp plot and some of the conflict measures defined in Gåsemyr and Natvig (2009). This indicates how the conflict expressed visually in the graphical plots can be soundly assessed numerically.