Decision Theoretic Aspects of Dependent Nonparametric Processes
Steven N. MacEachern · 2000
SUMMARY Nonparametric Bayesian methods have proven to be extremely useful for providing flexible models that are capable of fitting an extraordinarily wide array of data sets. Two of their most natural uses are in providing distributions for random effects and in providing large classes of models that elaborate on a parametric model. These models are appropriate for a great many data sets, allowing one, to write, for example, a regression model in which the mean of the response is a linear function of the covariate but where the error distribution can assume a nonparametric form. The flexibility brought about by use of an arbitrary error distribution is essential when any aspect of the analysis relies on the distribution of an individual case. Two common inferences that rely on this distribution are prediction about an individual case and investigation of whether an observation is outlying. An inadequacy of current nonparametric models is that they do not fully accomodate covariates. Currently, two distinct distributions of random effects are either identical or they are conditionally independent realizations from a nonparametric prior distribution. In the context of the regression problem, this limitation implies that the conditional distribution of the response is either restricted to lie in a location family, or (roughly) that the distribution of the response given covariate is not continuous in the covariate. These limitations are clearly much more restrictive than one would like. The remedy for these, and many other modelling inadequacies, lies in dependent nonparametric processes. In particular, extension of the Dirichlet process provides a class of models that are attractive conceptually and computationally, and that capture many fundamental modelling strategies which have heretofore been inaccessible. These models allow one to capture the notion of a response distribution which is, at each fixed level of the covariate, nonparametric, and which also provides continuity of the response distribution as a function of the covariate. Since these models allow for response distributions which “evolve” with the covariate, they facilitate inference about individual cases, allowing one to have predictive distributions of different shape for different levels of the covariate. Along with these new models come seemingly new inferential issues. This work investigates a key inferential issue that arises when dependent nonparametric processes are used as a component in a linear model. Although one can write nonparametric models that preserve the linear relationship between the mean of the response variable and the covariates,