We presented exact solutions and iteratively convergent approximations for inferences in Bayesian models associated with finitely generated convex sets of probability distributions. Almost all classes of distributions used in robust Statistics were reduced to the class of finitely generated convex sets.
Robust Bayesian analysis is a field with clear practical relevance. Any application of Bayesian networks must determine the relationship between the accuracy of probability values and the accuracy of inferences. Yet research on Bayesian networks has not fully explored this aspect of inference, due to the lack of algorithms for inferences with sets of distributions. This paper offers a solution to this problem, opening the field of robust Bayesian Statistics to graphical approaches.
There are several trade-offs that can be exercised when deciding which algorithm to use. Typically, a network is built or estimated, and then some critical variables and links are analyzed with respect to robustness. This process will rarely involve the consideration of too many credal sets at a time; more often, different portions of the network and structural assumptions will be analyzed carefully and separately. Exact algorithms may be feasible in most such cases where a few credal sets are under consideration.
Even when a small number of credal sets are present, there may be difficulties in running exact algorithms if each one of the credal sets is represented by a large number of vertices. In some cases, a representation in terms of linear inequalities may be much more compact than the enumeration of vertices. These cases are best handled with Lavine's algorithm and linear programming.
Finally, for the situations where many large credal sets are present, the QEM algorithm is indicated as it is the first algorithm for numeric inferences in Quasi-Bayesian networks with provably convergent behavior.
Tue Jan 21 15:59:56 EST 1997