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## QUASI-BAYESIAN THEORY AND POLYTOPIC CREDAL SETS

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) = infp isinK p(A), p(A) = supp isinK p(A).

In the remainder of this paper, we will refer either to maximization or minimization procedures; lower and upper bounds are closely related through the expression p(A) = 1 - p(Ac).

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

E[u] = infp isinK Ep[u] E[u] = supp isinK Ep[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.

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