Bayesian Analysis of Constrained Parameter and Truncated Data Problems Using Gibbs Sampling

Abstract

Constrained parameter problems arise in a wide variety of applications, including bioassay, actuarial graduation, ordinal categorical data, response surfaces, reliability development testing, and variance component models. Truncated data problems arise naturally in survival and failure time studies, ordinal data models, and categorical data studies aimed at uncovering underlying continuous distributions. In many applications both parameter constraints and data truncation are present. The statistical literature on such problems is very extensive, reflecting both the problems’ widespread occurrence in applications and the methodological challenges that they pose. However, it is striking that so little of this applied and theoretical literature involves a parametric Bayesian perspective. From a technical viewpoint, this perhaps is not difficult to understand. The fundamental tool for Bayesian calculations in typical realistic models is (multidimensional) numerical integration, which often is problematic in unconstrained contexts and can be well-nigh impossible for the kinds of constrained problems we consider. In this article we show that Bayesian calculations can be implemented routinely for constrained parameter and truncated data problems by means of the Gibbs sampler. Specific models discussed include constrained multinormal parameters, constrained linear model parameters, ordered parameters in experimental family models, data and order restricted parameters from exponential distributions, straight line regression with censoring and bivariate grouped data models. Analysis of data sets illustrating the first two of these settings is provided.