**Fabio Cozman**

This work is an upgraded version of a technical report at the School of Computer Science, Carnegie Mellon University (CMU-RI-TR 97-24).

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.

- The Basics of the Decision Model: Acts, States, Losses and Utilities
- Foundation of the Theory of Sets of Probabilities
- The Mathematical Axioms of Quasi-Bayesian Theory
- Important Definitions: Conditional Preferences and Independence
- Lower Expectations and Lower Previsions
- Lower Envelopes (or Coherent Lower Probabilities)
- Lower Probability, Choquet Capacities and Belief Functions
- Decision Making with Sets of Probabilities
- Bibliography
- About this document ...