Bayes-Invariant Transformations of Uncertainty Representations

摘要

Much effort has been expended on devising "conversions" of one uncertainty representation scheme to another- fuzzy to probabilistic, Dempster-Shafer to probabilistic, to fuzzy, etc. Such efforts have been hindered by the fact that uncertainty representation formalisms vary considerably in the degree of complexity of information which they encode. For example, 2~M - 1 numbers are required to specify a Dempster-Shafer basic mass assignment (b.m.a.) on a space with M elements; whereas only M - 1 numbers are required to specify a probability distribution on the same space. Consequently, any conversion of b.m.a.'s to probability distributions will result in a huge loss of information. In addition, conversion from one uncertainty representation formalism to another should be consistent with the data fusion methodologies intrinsic to these formalisms. For b.m.a.'s to be consistently converted to fuzzy membership functions, for example, Dempster's combination should be transformed into fuzzy conjunction in some sense. In this paper we show that a path out of such quandaries is to realize that in many applications all information must ultimately be reduced to state estimates and covariances. Adopting a Bayesian approach, we identify Bayes-invariant conversions between various uncertainty representation formalisms.