Hierarchical Bayesian Nonparametric Models with Applications∗

Hierarchical modeling is a fundamental concept in Bayesian statistics. The basic idea is that parameters are endowed with distributions which may themselves introduce new parameters, and this construction recurses. A common motif in hierarchical modeling is that of the conditionally independent hierarchy, in which a set of parameters are coupled by making their distributions depend on a shared underlying parameter. These distributions are often taken to be identical, based on an assertion of exchangeability and an appeal to de Finetti’s theorem. Hierarchies help to unify statistics, providing a Bayesian interpretation of frequentist concepts such as shrinkage and random effects. Hierarchies also provide ways to specify non-standard distributional forms, obtained as integrals over underlying parameters. They play a role in computational practice in the guise of variable augmentation. These advantages are well appreciated in the world of parametric modeling, and few Bayesian parametric modelers fail to make use of some aspect of hierarchical modeling in their work. ∗To appear in: N. Hjort, C. Holmes, P. Müller, & S. Walker (Eds.), Bayesian Nonparametrics in Practice, Cambridge, UK: Cambridge University Press.