A DUALITY PROPERTY FOR BAYES RULES WITH APPLICATIONS.
SOUTHERN METHODIST UNIV DALLAS TEX DEPT OF STATISTICS
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Many minimax rules are also solutions to a related Bayes or extended Bayes problem. This paper investigates the inverse to the above result which is that many Bayes or extended Bayes rules are also solutions to a related minimax problem. This relationship has been given herein the name duality property for Bayes rules. Interpretations and applications of the duality property are discussed. For example the duality property provides an objective justification for Bayes rules in that the dual problem does not depend upon a prior distribution. This is important from a military viewpoint because some problems such as threat classification of enemy radars and final attack of enemy submarines can be formulated very easily within the Bayesian framework. Minimax rules for several problems are presented to illustrate the obvious application to solving minimax problems. Minimax rules are useful in making decisions on tactics in order to minimize the maximum expected loss for a limited war or counterinsurgency situation. The loss function for the dual minimax problem in parametric form is defined here as a least favorable loss function because it plays a similar role to the least favorable distribution. The least favorable loss function notion is applied to reliability test and reconnaissance sensor designs to illustrate how this concept can be used to introduce prior information into a minimax problem. Least favorable loss functions for several well known problems are included for reference. Author
- Statistics and Probability