Convex sets of distributions

 

We consider theories that use convex sets of distributions to represent beliefs and to evaluate decisions. The set of distributions maintained by an agent is called the credal set [15]. 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.

Given a credal set K, a probability interval can be created for every event A by defining lower and upper bounds, called the lower and upper envelopes:

p(A) = inf_{p isinK} p(A) p(A) = sup_{p isinK} p(A).

We say that a probability distribution p() dominates a lower envelope p() if p(A) gep(A) for every event A.

We can also define an expected utility interval for every utility function u():

E[u] = inf_{p isinK} E_p[u] E[u] = sup_{p isinK} E_p[u]

Since utility functions induce expected utility intervals, it may be the case that decisions are incomparable (the ordering of possible decisions is a partial order) [15].

The upper envelopes and expectations can be obtained from the lower envelopes and expectations respectively. We have p(A) = 1 - p(A^c) and E[u] = - E[-u] for any event A and utility u().

Convex sets of distributions are interesting for several reasons, ranging from mathematical elegance to practical considerations of robustness (for an extensive discussion of this topic, consult Walley [25]). One of the common justifications is that assumptions of Bayesian theory are too strict: how can a real agent be required to specify a single number when explaining beliefs?