Bayesian Approximation and Invariance of Bayesian Belief Functions

1 I n t r o d u c t i o n The Dempster-Shafer theory is quite popular in knowledge based applications. However, it's exponential computational complexity is a stumbling block. Several researchers worked on the problem of reducing the computational burden of the theory. The work in this direction was initiated by Barnett [1]. The approach of reducing the number of focal elements by certain approximation scheme was taken by Voorbraak [14], Dubois and Prade [2], and Tessem [13]. The work on propagation of belief in networks can be found in Gordon and Shortliffe [3], Shenoy and Sharer [10], Sharer and Logan [8], Sharer, Shenoy and Mellouli [9], Kohlas and Monney [6]. Kennes and Smets presented fast algorithms using mSbius transforms [4, 5] and Wilson [15] gave a Monte-Carlo algorithm for belief computation. All these methods have given rise to efficient implementation of Dempster's combination rule. Smets Ill, 12] considered pignistic probability distribution based on belief function describing credal state for decision making. He arrived at this distribution based on axiomatic justification for generalized insufficient principle. In this paper, we present results on invariance of Bayesian belief functions. These results help us to understand and interpret Bayesian approximation from a new perspective. In the light of this interpretation, we show that the properties of Bayesian approximation follow directly from the properties of the combination operator @ of Dempster's combination rule. Further, given these set of properties, Bayesian approximation is unique in the class of approximations which can be obtained as a combination of Bayesian belief function and any other belief function. Finally, we show that Bayesian approximation has some limitations. Due to restrictions on number of pages, proofs of the theorem and corollaries are not included in this paper.