@bullet @bullet @bullet @bullet Bayesian Case Studies in Nonparametrics

Elements of Bayesian nonparametric statistical thought are explored in a series of case studies. Interpretation of a measurement as continuous, ordered, polychotomous, or dichotomous provides a framework in which examples are presented. Bayesian analogues to frequentist nonparametrics and overt Bayesian techniques are employed. Examples included are as follows: (1) averaging over families of distributions, (2) estimation of a single distribution function, (3) comparing several distribution functions, (4) estimating the coefficient of a concomitant variable affecting a distribution function, (5) monitoring compliance with a dichotomous measurement, and (6) using the multinomial for a categorization of any measurement's range. Lindley (1972, §12.2) provides an intitial sketch. Hill's (1968) nonparametric Bayesian construct and Berliner and Hill's (1988) application to survival are also reviewed. A commonality in the mechanics of these examples is the calculation of a marginal distribution over model parameters. Many are predictive distributions, resulting from an average over a likelihood and vague prior, and leaving observables for the calculations, as described by Roberts (1965) and advocated by Geisser (1971). Other specific observations from these efforts include the following points: (1) nonparametrics is typified by a proliferation of parameters; (2) biased estimation may provide smaller mean squared errors; (3) point null hypotheses are intrinsically unlikely; (4) the practical aspects of conditional inference with proportional hazards are embraced by Bayesian considerations; and (5) Fisher's Exact Test is a predictive distribution.