Let \(\Theta\) be a Polish space with Borel \(\sigma\)-algebra \({\mathcal B}(\Theta)\) and let X be a convex, compact, metrizable subset of a locally convex topological vector space with Borel \(\sigma\)-algebra \({\mathcal B}(X)\). Let \(\mu\) be a probability measure on (X,\({\mathcal B}(X))\) and let \(\Gamma\) be a map taking points in X to nonempty, closed subsets of \(\Theta\). Given this structure, the author undertakes the study of suitably defined, for each \(A\subset \Theta\), belief functions BEL(A) and plausibility functions PL(A). The author gives an intuitive explanation as follows:
Draw x at random according to \(\mu\). Then BEL(A) is the probability that the random set \(\Gamma\) (x) is contained in A and PL(A) is the probability that the random set \(\Gamma\) (x) hits A. Moreover \(BEL(\Phi)=PL(\Phi)=0\), \(BEL(\Theta)=PL(\Theta)=1\) and BEL(A)\(\leq PL(A)\), with equality iff BEL is a probability measure. BEL and PL may be thought of as the lower and upper bounds of a class of probability measures.
It is claimed that the mathematical structure of belief functions makes them suitable for generating classes of prior distributions to be used in robust Bayesian inference. In particular, the problem of finding extrema over the set of priors is reduced to that of maximizing and minimizing the likelihood over sets in the parameter space followed by an integration. The author acknowledges that “this is not an easy task” and that “much remains to be done from a practical point of view”.