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: (C_j=1^m p_1,j ). In this case the set of distributions modeled by the network is:

( C_j=1^m p_1,j ) prod_i>1 p_i.

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

r_j(x) = p_1,j prod_i>1 p_i.

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:

  C_j,k ( p_1,j p_2,k ) prod_i>2 p_i.

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 p_1,j and p_2,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.