Nonparametric Modeling of Hierarchically Exchangeable Data

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. Some key words: nonparametric Bayes, logistic normal, multivariate normal, multinomial, Dirichlet distribution, repeated measures.