Empirical Bayes Methods

J. S. Maritz, T. Lwin

• Publisher: Taylor & Francis
• Year: 2018
• Print ISBN: 9780815350286
• eBook ISBN: 9781351140621
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

Originally published in 1970; with a second edition in 1989. Empirical Bayes methods use some of the apparatus of the pure Bayes approach, but an actual prior distribution is assumed to generate the data sequence. It can be estimated thus producing empirical Bayes estimates or decision rules.

In this second edition, details are provided of the derivation and the performance of empirical Bayes rules for a variety of special models. Attention is given to the problem of assessing the goodness of an empirical Bayes estimator for a given set of prior data. Chapters also focus on alternatives to the empirical Bayes approach and actual applications of empirical Bayes methods.

CHAPTER 1

Introduction to Bayes and empirical Bayes methods

1.1 The problem, Bayes conventional and empirical Bayes methods

Empirical Bayes (EB) and related techniques come into play when data are generated by repeated execution of the same type of random experiment. The individual experiments are often called component experiments, or simply components. It is convenient when considering the data from a particular component to think of it as the current component which has been preceded by the other components. Empirical Bayes methods provide a way in which such historical data can be used in the assessment of the current results. This temporal view of the data sequence is a convenience and does not play an active role in EB analysis.

An early example of an EB nature is given by von Mises (1943); see also Chapter 8, section 8.3.1. In examining the quality of a batch of water for possible bacterial contamination m = 5 samples of a given volume are taken. A sample registers a positive result if it contains at least one bacterium. Interest centres on the probability, 0, of a positive result. Typically there are many repetitions of this experiment with different batches, and the probability θ can be regarded as varying randomly between experiments according to a prior distribution G(θ). For a given θ the probability of x positive results in m = 5 samples is

p(x|θ)=(5x)θx(1−θ)5−x,

and in repetitions of the same procedure with different batches the marginal distribution of the number of positive results in five samples is the mixed binomial distribution

pG(x)=∫(5x)θx(1−θ)5−xdG(θ).

If the distribution G is known, a Bayesian analysis of the current experiment can be performed. For example, a Bayes point estimate of θ can be calculated as a possible competitor for the classical maximum likelihood estimate. When the prior, or mixing, distribution G is not known it is possible to estimate it by using the observed marginal distribution of the x values. The essence of the EB method in this case is that all calculations of a Bayes nature are performed after replacing G by its estimate. In the example discussed by von Mises there are N = 3420 observations f...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Acknowledgements
• Preface
• Notation and abbreviations
• 1 Introduction to Bayes and empirical Bayes methods
• 2 Estimation of the prior distribution
• 3 Empirical Bayes point estimation
• 4 Empirical Bayes point estimation: vector parameters
• 5 Testing of hypotheses
• 6 Bayes and empirical Bayes interval estimation
• 7 Alternatives to empirical Bayes
• 8 Applications of EB methods
• References
• Author index
• Subject index