Robust Bayesianism: Relation to Evidence Theory

We are interested in understanding the relationship between Bayesian inference and evidence theory. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of Dempster-Shafer’s evidence theory. We interpret imprecise probabilities as imprecise posteriors obtainable from imprecise likelihoods and priors, both of which are convex sets that can be considered as evidence and represented with, e.g., DS-structures. Likelihoods and prior are in Bayesian analysis combined with a place’s parallel composition. The natural and simple robust combination operator makes all pairwise combinations of elements from the two sets representing prior and likelihood. Our proposed combination operator is unique, and it has interesting normative and factual properties. We compare its behavior with other proposed fusion rules, and earlier efforts to reconcile Bayesian analysis and evidence theory. The behavior of the robust rule is consistent with the behavior of Fixsen/Mahler’s modified Dempster’s (MDS) rule, but not with Dempster’s rule. The Bayesian framework is liberal in allowing all significant uncertainty concepts to be modeled and taken care of and is therefore a viable, but probably not the only, unifying structure that can be economically taught and in which alternative solutions can be modeled, compared and explained.