Identifying influential model choices in Bayesian hierarchical models

Real-world phenomena are frequently modelled by Bayesian hierarchical models. The buildingblocks in such models are the distribution of each variable conditional on parent and/or neighbour variables in the graph. The specifications of centre and spread of these conditional distributions may be well-motivated, while the tail specifications are often left to convenience. However, the posterior distribution of a parameter may depend strongly on such arbitrary tail specifications. This is not easily detected in complex models. In this paper we propose a graphical diagnostic which identifies such influential statistical modelling choices at the node level in any chain graph model. Our diagnostic, the local critique plot, examines local conflict between the information coming from the parents and neighbours (local prior) and from the children and co-parents (lifted likelihood). It identifies properties of the local prior and the lifted likelihood that are influential on the posterior density. We illustrate the use of the local critique plot with applications involving models of different levels of complexity. The local critique plot can be derived for all parameters in a chain graph model, and is easy to implement using the output of posterior sampling.