Bayes-like Decision Making with Upper and Lower Probabilities

Abstract

We consider the use of interval-valued probabilities to represent the support lent to the hypothesis that the parameter value θ lies in a subset A of the parameter set Θ when we observe x, know the likelihoods {f_Θ: θεΘ}, and have some prior information concerning the parameter. Our model for prior information is that of a salient prior distribution in which we have little confidence, although we have much less confidence in any alternative prior. We consider notions of acceptable and coherent decision making as well as notions of being able to achieve a Bayes rule and least commitment. Throughout we are motivated to preserve some of the elements of Bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge.

Key Words:

• Bayesian decision making
• Belief functions
• Lower envelopes
• Coherence
• Lower probabilities