Abstract
The paper discusses generalization of conditioning in Evidence theory proposed recently, and proposes a new generalized conditioning with an uncertain belief. It also shows that a conventional method of evidential reasoning could be interpreted by the proposed generalized conditioning. A conditioning rule proposed recently by Kimala et al. satisfies the following three requirements: a) focal elements of conditional basic belief assignment are limited to subsets of the given condition, b) conditional bba must be defined for any condition except for the empty set, and c) conditional bba with condition of the universal set equals to bba with no condition. All those requirements are natural and appropriate with understanding that Evidence theory deals with subjective uncertainty of humans, though there are no conditioning rules satisfying all of the three requirements. They also proposed a generalized conditioning that loosens a requirement a) by introducing reliability of the condition. The paper generalizes further Kimala et al.'s generalized conditioning. The proposed generalized conditioning uses a belief represented by a bba as the condition. Ichihashi et al. and Dubois et al. have already proposed similar conditioning rules. However, since they are extensions of Bayesian conditioning and Jeffrey's rule, they cannot satisfy the requirement b). The conditioning proposed in the paper satisfy both b) and c), though a) is lost because of the generalization. The paper also discusses the relation between a conventional evidential reasoning and conditioning rules. The evidential reasoning is understood as an extension of Bayesian reasoning. In the paper we show that it could be interpreted as a combination of evidence in Transferable Belief Model proposed by Smets et al., as well as the proposed generalized conditioning rule with an uncertain belief.
