Belief representation for counts in Bayesian inference and experimental design

Abstract

Bayesian inference for such things as collections of related binomial or Poisson distributions typically involves rather indirect prior specifications and intensive numerical methods (usually Markov chain Monte Carlo) for posterior evaluations. As well as requiring some rather unnatural prior judgements this creates practical difficulties in problems such as experimental design. This thesis investigates some possible alternative approaches to this problem with the aims of making prior specification more feasible and making the calculations necessary for updating beliefs or for designing experiments less demanding, while maintaining coherence. Both fully Bayesian and Bayes linear approaches are considered initially. The most promising utilises Bayes linear kinematics in which simple conjugate specifications for individual counts are linked through a Bayes linear belief structure. Intensive numerical methods are not required. The use of transformations of the binomial and Poisson parameters is proposed. The approach is illustrated in two examples from reliability analysis, one involving Poisson counts of failures, the other involving binomial counts in an analysis of failure times. A survival example based on a piecewise constant hazards model is also investigated. Applying this approach to the design of experiments greatly reduces the computational burden when compared to standard fully Bayesian approaches and the problem can be solved without the need for intensive numerical methods. The method is illustrated using two examples, one based on usability testing and the other on bioassay.