Quasi-Bayesian theory

 

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) = inf_{p is in K} p(A)

p(A) = sup_{p is in K} p(A) .

In the remainder of this paper, we will refer to maximization procedures for the determination of upper bounds, and leave implicit that similar minimization procedures lead to lower bounds. It is also possible to obtain lower bounds from upper bounds using the general identity p(A) = 1 - p(A^c) for any event 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.