**Attention: in the HTML version, the overline that indicates
upper bounds is modified to a strike-out markup (since overlines are
not available in HTML yet.**

Quasi-Bayesian theory uses convex sets of distributions to represent beliefs and to evaluate decisions [Giron & Rios1980]. Several other theories use probability intervals: inner/outer measures [Good1983, Halpern & Fagin1992, Ruspini1987, Suppes1974], lower probability theory [Breese & Fertig1991, Chrisman1996b, Fine1988, Smith1961]), convex Bayesianism [Kyburg Jr.1987], Dempster-Shafer theory [Ruspini1987, Shafer1987], probability/utility sets [Seidenfeld1993].

Given a set of functions f_{i}, their convex combination
is determined by sum_{i} alpha_{i} f_{i}, where all alpha_{i} are positive
and sumalpha_{i} = 1. The convex combination of the elements of a
set generates a convex set, which can be expressed as the convex hull
of its vertices.
We denote the convex hull of a finite set of functions f_{i}()
by _{i} f_{i}.

The set of distributions maintained by an agent is called the *credal*
set, and its existence is postulated on the grounds of axioms about
preferences [Giron & Rios1980]. 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. An
introduction to technical aspects of
Quasi-Bayesian theory, with a larger list of references, can be found at
http://www.cs.cmu.edu/~fgcozman/qBayes.html.

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 [Giron & Rios1980]. Second, to obtain maximum and minimum values of posterior probabilities, we must look only at the vertices of the posterior credal sets [Walley1991].

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 is in K} p(A)

~~p~~(A) = _{p is in K} p(A) .

Notice that a convex set of distributions always creates lower and upper
bounds of probability, but it is *not* true that a set of lower and
upper bounds of probability always defines a unique convex set
of distributions.
Much of the previous work on interval analysis of probability has
foundered due to this fact. Application of Bayes rule essentially demands the
construction of a convex set that obeys original prior intervals, but
this construction is mathematically ill-defined. The Quasi-Bayesian approach
sidesteps this difficulty by taking convex sets as basic entities.

Tue Jan 21 15:59:56 EST 1997