We consider theories that use convex sets of distributions to
represent beliefs and to evaluate decisions.
The set of distributions maintained by an agent is called the
*credal* set [15]. To simplify terminology, we use the
term credal set only when it refers to a set of distributions containing
more than one element. Convex sets of conditional distributions are used
to represent conditional beliefs. Inference is performed by applying Bayes
rule to each distribution in a prior credal set; the posterior credal
set is the union of all posterior distributions.

Given a credal set K, a probability interval can be created for every event A by defining lower and upper bounds, called the lower and upper envelopes:

__p__(A) = _{p isinK} p(A)
_{p isinK} p(A).

We can also define an expected utility interval for every utility function u():

__E__[u] = _{p isinK} E_{p}[u] _{p isinK} E_{p}[u]

The upper envelopes and expectations can be obtained from the
lower envelopes and expectations respectively. We have
__p__(A^{c}) and
__E__[-u] for any event A
and utility u().

Convex sets of distributions are interesting for several reasons, ranging from mathematical elegance to practical considerations of robustness (for an extensive discussion of this topic, consult Walley [25]). One of the common justifications is that assumptions of Bayesian theory are too strict: how can a real agent be required to specify a single number when explaining beliefs?

© Fabio Cozman[Send Mail?]

Sun Jun 29 22:16:40 EDT 1997