Introduction to the Theory of Sets of Probabilities
(Probability Intervals, Belief Functions,
Lower Probability, Lower Expectations, Choquet Capacities,
Robust Bayesian Methods, etc...)
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
There are two main pieces of information off of this page:
There are two other sources of material on these things
in the web (you can get more recent references through them):
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 . Their
paper is very nice; they call the resulting theory Quasi-Bayesian
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;
send me e-mail
with a pointer to your work (or other
work that you think is relevant). Thanks.
of the content that you can reach from this page.
Why so many words in the sub-title of this page?
There are several similar generalization of probability that use
sets of probability distributions:
These theories have points of divergence, but this work tries to
emphasize the points where there is agreement.
- 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
or Choquet Capacities. Special cases of such structures are
the Monotone Choquet Capacities and the Lower Envelopes.
Infinitely Monotone Choquet Capacities are sometimes called
Such structures can be in most cases represented by convex sets of probability
- 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.
My work with sets of probabilities
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
Most of the results in the second paper,
and a summary of the third paper, can be found at
My interests in the theory of sets of probability follow
some general lines. First, I'm interested in efficient algorithms
to obtain posterior quantities.
I'm now trying to extend such algorithms to deal with more
complicated cases; for example, situations where observations
may have probability zero, and situations where judgements of
independence are stated. Second, I'm interested in
concepts and properties of irrelevance/independence connected
to the theory of sets of probabilities. A starting point
- F. G. Cozman.
Credal networks, Artificial Intelligence Journal,
vol. 120, pp. 199-233, 2000.
(This paper is a mature version of papers presented at
- 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.
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
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:
These books are very dense and require some background.
I'm trying to construct these informal pages for the reader that is not
entirely familiar with the
theory of sets of probabilities,
but has already learned some probability and decision theory.
- I. Levi, whose The Enterprise of Knowledge 
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
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:
J. O. Berger.
Statistical Decision Theory and Bayesian Analysis.
F. J. Giron and S. Rios.
Quasi-Bayesian behaviour: A more realistic approach to decision
In J. M. Bernardo, J. H. DeGroot, D. V. Lindley, and A. F. M. Smith,
editors, Bayesian Statistics, pages 17-38. University Press, Valencia,
The Enterprise of Knowledge.
The MIT Press, Cambridge, Massachusetts, 1980.
Statistical Reasoning with Imprecise Probabilities.
Chapman and Hall, New York, 1991.
Thanks for the visit; you're visitor
since July 15, 1996.
I typed almost all of this document with gnu-emacs using LaTeX commands.
The LaTeX documents were converted to postscript with
dvips, and to HTML with
The translations from LaTeX to postscript and LaTeX to LaTeX2HTML
were coordinated by some simple makefiles.
I drew the figures in CorelDraw (in Windows), and used xv (in Unix) to
help me produce the imagemaps (wow, that was quite a pain).
© Fabio Cozman[Send Mail?]
This work started at the Robotics Institute at the School of
Computer Science, Carnegie Mellon University. I had
a scholarship from CNPq (Brazil). Thanks to these organizations!