TOWARD A RECONCILIATION OF THE BAYESIAN AND FREQUENTIST APPROACHES TOPOINT ESTIMATION

The Bayesian and frequentist approaches to point estimation are review ed. The status of the debate regarding the use of one approach over th e other is discussed, and its inconclusive character is noted. A crite rion for comparing Bayesian and frequentist estimators within a given experimental framework is proposed. The competition between a Bayesian and a frequentist is viewed as a contest with the following component s: a random observable, a true prior distribution unknown to both stat isticians, an operational prior used by the Bayesian, a fixed frequent ist rule used by the frequentist, and a fixed loss criterion. This com petition is studied in the context of exponential families, conjugate priors, and squared error loss. The class of operational priors that y ield Bayes estimators superior to the ''best'' frequentist estimator i s characterized. The implications of the existence of a threshold sepa rating the space of operational priors into good and bad priors are ex plored, and their relevance in areas such as Bayesian robustness and t he elicitation of prior distributions is discussed. Both the theoretic al and empirical results presented in this article suggest that the me thod to be favored in a particular application depends crucially on th e quality of the prior information available, with Bayesian and freque ntist methods each emerging as preferable under specific, and compleme ntary, circumstances.