Contributions to Bayesian statistical analysis: Model specification and nonparametric inference.

摘要

This dissertation concerns two topics in Bayesian statistical analysis: model specification and nonparametric inference. We begin, following de Finetti, with (i) the observation that perhaps the most general way to define the phrase Bayesian model is by regarding such a model as a joint predictive distribution p(y) for observables (data) y that have not yet been observed, and with (ii) the principle that in specifying p( y) we would like the modeling process to be driven as much as possible by the context of the real-world problem at issue. In simple situations with discrete outcomes this leads via de Finetti's notion of exchangeability to parametric models (those in which the underlying mechanism generating the observed data is taken to be a member of a standard family of probability distributions indexed by a parameter vector); if instead the outcomes are continuous, exchangeability considerations arising from problem context lead to nonparametric models (those in which uncertainty about the underlying data-generating mechanism is modeled by placing probability distributions on the space of all possible cumulative distribution functions).; After an introduction to the basic issues in Chapter 1, the second chapter of the dissertation identifies two types of tools needed in Bayesian parametric modeling: (a) given two candidate models i1 and M 2 under consideration, investigators need a method for choosing between them, and (b) having made a series of comparisons among the k models under consideration using methods of the type identified in step (a), and having identified the best model M* of the k under consideration, investigators need a method for deciding whether M* is good enough to stop looking for even better models.; In Chapter 2 we argue that model selection is really a decision problem which should be approached by maximizing expected utility, and this perspective leads us to examine the performance of a particular tool of type (a), the predictive log-score (LS) criterion, based on a utility structure that rewards accuracy in predicting future data.; In Chapter 3 we undertake a simulation study to explore the ability of Bayesian parametric and nonparametric models to provide an adequate fit to count data, of the type that would routinely be modeled parametrically either through fixed effects or random-effects Poisson regression.; Chapter 4 presents new Bayesian nonparametric methodology for quantile regression, which is useful in situations where it is more flexible to quantify the relationship between the response variable and available covariates through regression on a set of quantiles of the response distribution, rather than just on the mean, as in traditional regression models. (Abstract shortened by UMI.)