The Selection of Prior Distributions by Formal Rules

Abstract

Subjectivism has become the dominant philosophical foundation for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called “noninformative” priors, that is, priors constructed by some formal rule. We review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used. We give special emphasis to Jeffreys's rules and discuss the evolution of his viewpoint about the interpretation of priors, away from unique representation of ignorance toward the notion that they should be chosen by convention. We conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly: When sample sizes are small (relative to the number of parameters being estimated), it is dangerous to put faith in any “default” solution; but when asymptotics take over, Jeffreys's rules and their variants remain reasonable choices. We also provide an annotated bibliography.

Key Words:

• Bayes factors
• Entropy
• Haar measure
• Improper priors
• Jeffreys's prior
• Marginalization paradoxes
• Noninformative priors
• Reference priors