Fabio Cozman
This is an attempt to briefly cover essential aspects of the theory of sets of probabilities and its cousins: lower previsions, lower probability, lower envelopes and Choquet capacities. An even quicker exposition of these theories and the relations among them is available. All these theories deal with sets of probability distributions; they augment/enrich/generalize/improve (pick your word) the infra-structure of usual Bayesian decision theory.
I use, as a starting point, the axiomatization of the theory proposed by Giron and Rios [11], which they call Quasi-Bayesian theory. This formulation is simple and general; other theories can easily be derived or explained from it. Their name emphasizes similarities with standard Bayesian theory; some view the theory of sets of probabilities as quite different from Bayesian theory and would not adopt Giron and Rios suggestion. Also, it must be noted that the original Quasi-Bayesian theory by Giron and Rios was quite elegant but did not include discussions of conditionalization and independence, and they did not have a clear statement of decision criteria. This report presents a more complete theory by filling the gaps with ideas that have been proposed in a variety of contexts in the last decade. The goal of this report is to present the theory in a unified, informal format so that its scope can be easily appreciated.