**Fabio Cozman**

What I hope is that these pages contain a brief but reasonably general presentation of the foundations of theories that handle sets of probability distributions. There are many such theories: Quasi-Bayesian theory, Lower Probability, Lower Expectations, Choquet Capacities, Robust Bayesian Methods, and some other similar theories. I feel sorry that I can't possibly refer to all the good work that has been published on these topics --- I attempted to refer to some representative papers and books, mostly of foundational character, and most of them have been written prior to 1993.

There are two main pieces of information off of this page:

- The one-page Theory in a nutshell.
- A longer, but still quite informal, description of the theory of sets of distributions (and related theories).

There are two other sources of material on these things in the web (you can get more recent references through them):

- The web site for the First Symposium on Imprecise Probabilities and Their Applications (which I co-organized).
- The Imprecise Probabilities Project (which I co-edit).

Most of the content in this web site talks about foundational concepts and basic results. A collection of practical results would also be useful, but I think the first step must be to present a consistent theory.

I focus only on proposals that maintain the basic infra-structure of Bayesian theory and augment/enrich/generalize it. Proposals that require an entirely different view of uncertainty (like Dempster-Shafer theory), or which handle other concepts (like fuzzy logic) are not covered here. Among all the possible theories that use sets of probability distributions to represent uncertainty, there is a particular axiomatization that is very simple to present and understand. It is the axiomatization given by two statisticians, Giron and Rios, in 1980 [2]. Their paper is very nice; they call the resulting theory Quasi-Bayesian theory.

The original theory by Giron and Rios was quite elegant but did not include discussions of conditionalization and independence; they also did not have a clear statement of decision criteria. I try to present their theory and fill in those gaps with ideas that have been proposed in a variety of contexts in the last decade; the goal is to present the theory in a unified format so that its scope can be better analyzed.

I'm aware that there is a lot of excellent work that I have not reviewed; please send me e-mail with a pointer to your work (or other work that you think is relevant). Thanks.

There are postscript versions of the content that you can reach from this page.

- The theories of
*Lower Expectations*and*Lower Previsions*use intervals of expected losses to generate sets of distributions. - A slightly different approach in theories that impose
axioms on events. The
resulting structures are generalizations of probability called
*Lower Probabilities*or*Choquet Capacities*. Special cases of such structures are the*Monotone Choquet Capacities*and the*Lower Envelopes*. Infinitely Monotone Choquet Capacities are sometimes called*Belief functions*. Such structures can be in most cases represented by convex sets of probability distributions. - From a slightly different perspective, many statisticians use sets of distributions to study the robustness of a statistical analysis.

I hope these pages are useful for anyone interested in sets of distributions, but because I work more in Robotics and Artificial Intelligence, I can better understand the theory from this point of view.

I work both on foundational and algorithmic issues (with emphasis on the later). Most of my work on this theory can be grasped through the papers

- F. G. Cozman.
Credal networks,
*Artificial Intelligence Journal*, vol. 120, pp. 199-233, 2000. (This paper is a mature version of papers presented at UAI97 and UAI98.) - F. G. Cozman.
Computing posterior upper expectations,
*International Journal of Approximate Reasoning*, vol. 24, pp. 191-205, 2000. - F. G. Cozman.
Calculation of Posterior Bounds Given Convex Sets of
Prior Probability Measures and Likelihood Functions,
*Journal of Computational and Graphical Statistics*, vol. 8(4), pp. 824-838, 1999.

- F. G. Cozman.
*Computing Posterior Upper Expectations*, First International Symposium on Imprecise Probabilities and Their Applications (ISIPTA), pp. 131-140, Ghent, Belgium, June/July, 1999.

- F. G. Cozman.
Separation
Properties of Sets of Probability Measures.
*XVI Conference on Uncertainty in Artificial Intelligence*, pp. 107-115, San Francisco, California, July 2000. - F. G. Cozman.
Irrelevance and Independence Axioms in Quasi-Bayesian Theory,
*European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU)*, London, England, published in Symbolic and Quantitative Approaches to Reasoning with Uncertainty, A. Hunter e S. Parsons (eds.), pp. 128-136, Springer, July, 1999.

While working with these things, I have developed algorithms for the JavaBayes system, where I explore robust inferences with Quasi-Bayesian networks, using both local and global perturbations.

I have also pursued some different directions, looking at the problem of sequential-decision making associated with observations, and also exploring the possibility of learning convex sets of probability from data.

Most of the foundational issues in the theory of sets of probabilities can be absorbed through the work of two researchers:

- I. Levi, whose
*The Enterprise of Knowledge*[3] is a great analysis of many philosophical issues related to the theory. If you want Philosophy, you probably want to read this. - P. Walley, whose
*Statistical Reasoning with Imprecise Probabilities*[4] is a tremendous summary of all that has been said about the theory in the field of Statistics (and also tries some connections with Economics and Artificial Intelligence).

Here are four references that capture a vast portion of the theory of sets of probabilities:

**1**-
J. O. Berger.
*Statistical Decision Theory and Bayesian Analysis*. Springer-Verlag, 1985. **2**-
F. J. Giron and S. Rios.
Quasi-Bayesian behaviour: A more realistic approach to decision
making?
In J. M. Bernardo, J. H. DeGroot, D. V. Lindley, and A. F. M. Smith,
editors,
*Bayesian Statistics*, pages 17-38. University Press, Valencia, Spain, 1980. **3**-
I. Levi.
*The Enterprise of Knowledge*. The MIT Press, Cambridge, Massachusetts, 1980. **4**-
P. Walley.
*Statistical Reasoning with Imprecise Probabilities*. Chapman and Hall, New York, 1991.

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