Convex Bayes Decision Theory

The basic concepts of Levi's epistemic utility theory and credal convexity are presented. Epistemic utility, in addition to penalizing error as is done with traditional Bayesian decision methodology, permits a unit of informational value to be distributed among the hypotheses of a decision problem. Convex Bayes decision theory retains the conditioning structure of probability-based inference, but addresses many of the objections to Bayesian inference through relaxation of the requirement for numerically definite probabilities. The result is a decision methodology that stresses avoiding errors, and seeks decisions that are likely to be highly informative as well as true. By relaxing the mandatory requirement for unique decisions and point estimates in all cases, decision and estimation criteria are obtained that do not demand more than it is possible to obtain from the data, and permit a natural man-in-the-loop interface. Applications are provided to illustrate the theory.