The Bayesian approach to statistics : a review of methodology with selected applications : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University
In this thesis we present a review of the Bayesian approach to Statistical Inference. In Chapter One we develop the theory and methodology behind the approach. Starting from its basis in subjective probability we outline the Bayesian philosophy towards such problems as Point and Interval estimation, Hypothesis testing and Decision Theory. For each of these areas, we indicate the corresponding Classical approach and comment on the differences between this and the Bayesian one. We then develop the idea of conjugate families of prior distributions which is central to the practice of Bayesian statistics, and follow this with a section on the assessment of subjective probability distributions, their functional specification and the problem of mathematically
representing a state of 'ignorance'.
The Decision Theoretic approach to statistical analysis is then integrated into the Bayesian
framework, and reference is made to the assessment of 'loss' functions, and their subjective nature. Finally we consider the concepts of Empirical Bayes, Exchangeability, and Likelihood, and their relevence to the Bayesian scheme.
Chapter Two consists of a review of areas such as econometrics, medicine, industry, and education, where Bayesian methods have been applied, accompanied by a number of particularly interesting applications which illustrate the principles outlined in chapter one.