Hierarchically exchangeable data are characterized by the exchangeability of a population of units and the exchangeability of observations from each individual unit. A flexible model for such data is the hierarchical logistic-normal model, which provides unconstrained sampling distributions at the within-unit level and an unconstrained covariance structure at the betweenunit level. Also, the sampling distribution at the between-unit level is unimodal in a weak sense. Parameter estimation and inference for the hierarchical logistic-normal model is relatively straightforward via Markov chain Monte Carlo or an approximate EM algorithm. These and other features of the hierarchical logistic normal model are explored, and the model is applied to the analysis of tumor locations in a mammalian population. A comparison is made to a similar data analysis based on Dirichlet distributions.

Abstract: Hierarchical models are increasingly used in many applications. Along with this increase use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate in objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated. Key words and phrases: Model checking; model criticism; objective Bayesian methods; p-values. 1.

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. 1

A simulation-based goodness-of-fit test for

ABSTRACT. The present work focuses on extensions of the posterior predictive p-value (ppp-value) for models with hierarchical structure, designed for testing assumptions made on underlying processes. The ppp-values are popular as tools for model criticism, yet their lack of a common interpretation limit their practical use. We discuss different extensions of ppp-values to hierarchical models, allowing for discrepancy measures that can be used for checking properties of the model at all stages. Through analytical derivations and simulation studies on simple models we show that similar to the standard ppp-values, these extensions are typically far from uniformly distributed under the model assumptions and can give poor power in a hypothesis testing framework. We propose a calibration of the p-values, making the resulting calibrated p-values uniformly distributed under the model conditions. Illustrations are made through a real example of multinomial regression to age distributions of fish.

We compliment Bayarri and Castellanos (BC) on producing an interesting and insightful paper on model checking applied to the second level of hierarchical models. Distributions of test statistics (functions