Fusion of probabilistic evidence

Fusion refers to the combination of two or more probability assignments to pieces of evidence that support the same hypotheses. The probability assignments usually result from different inference paths in reasoning and are, in general, different. Given a set of probability assignments for evidence to be fused, it is well known that certain constraints, called consistent bounds, must be satisfied. These bounds arise from the theory of probability and define an admissible domain for the fused evidence. However, because the bounds are, in general, interactive, a general methodology for computing the admissible domain other than a brute-force numerical approach (linear programming) is lacking. This paper examines the role of interaction in evidence fusion and demonstrates the effect of interaction on the fused evidence. A simple case consisting of one hypothesis supported by two pieces of evidence is considered, and the interactive bounds and admissible domain are derived analytically. In particular, the effect of different dependency assumptions on the consistent bounds is derived to show that the assumption of conditional dependence can lead to inconsistencies under certain circumstances, that is, the fused evidence lies outside the admissible domain. Uncertain evidence expressed in the form of bounds is not very useful in practice because the bounds tend to become large as the uncertainty is cascaded from one level to another in inferencing. Point evidence may be more helpful. Three suggestions for obtaining consistent point estimates from the consistent bounds are presented, and numerical examples are given.