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Formal Rules for Selecting Prior Distributions: A Review and Annotated Bibliography

Robert E. Kass and Larry Wasserman

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 point of view 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 and phrases: Bayes factors, coherence, data-translated likelihoods, Entropy, Fisher information, Haar measure, improper priors, insufficient reason, Jeffreys's prior, marginalization paradoxes, noninformative priors, nuisance parameters, reference priors, sensitivity analysis.


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