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# Finitely generated convex sets of distributions

A finitely generated credal set is the convex hull of a finite number of probability distributions.

Consider the situation where a single variable x1 is associated with a credal set, and the credal set for x1 is the convex hull of m different distributions: (Cj=1m p1,j ). In this case the set of distributions modeled by the network is:

( Cj=1m p1,j ) prodi>1 pi.

This credal set is convex, with m vertices. The vertices are:

rj(x) = p1,j prodi>1 pi.

Consider a Quasi-Bayesian network with two credal sets associated with variables x1 and x2. To produce a convex set of joint distributions, we use the convex hull:

Cj,k ( p1,j p2,k ) prodi>2 pi.

The ``convexification'' of the joint set of distributions does not affect the ordering of decisions obtained by expected utility. Any decision or inference in a Quasi-Bayesian procedure is obtained through expectation, and expectation is maximized/minimized only at the vertices of the credal set. Since the vertices of this convex hull are the same distributions generated by cross-multiplication of terms p1,j and p2,k, we do not lose information by taking the convex hull operation.

This ``convexification'' technique is assumed in the remainder of the paper. When a network with several credal sets is used, we take the convex hull of the combined credal sets.