Quasi-Bayesian theory uses convex sets of distributions to represent beliefs and to evaluate decisions [16]. Several other theories use similar representations: inner/outer measures [17, 19, 30, 36], lower probability theory [4, 8, 15, 35]), convex Bayesianism [23], Dempster-Shafer theory [34], probability/utility sets [33].

The convex set of distributions maintained by an agent is called
the *credal* set, and its existence is postulated on the grounds
of axioms about preferences [16]. 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.

We use two well-known results about posterior credal sets in this paper. First, to obtain a posterior credal set, one has to apply Bayes rule only to the vertices of a prior credal set and take the convex hull of the resulting distributions [16]. Second, to obtain maximum and minimum values of posterior probabilities, we must look only at the vertices of the posterior credal sets [37].

Given a convex set K of probability distributions, a probability interval can be created for every event A by defining lower and upper bounds:

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

Lower and upper expectations for a function u(x) are defined as:

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

A credal set always creates lower and upper
bounds of probability, but a set of lower and
upper bounds of probability does not define a *unique* credal
set [37, section 2.7,].
The Quasi-Bayesian approach
sidesteps this difficulty by taking convex sets as basic entities.

A *polytopic credal set* is the convex
hull of a finite number of probability distributions, i.e., it is
a polytope in the space of all probability distributions.
As we assume that polytopic credal sets are specified over
local (presumably small) structures, we assume that representations
of polytopic credal sets in
terms of vertices and inequalities can be used interchangeably.

Fri May 30 15:55:18 EDT 1997