A sufficient condition for belief function construction from conditional belief functions

Abstract

It is commonly acknowledged that we need to accept and handle uncertainty when reasoning with real world data. The most profoundly studied measure of uncertainty is the probability. However, the general feeling is that probability cannot express all types of uncertainty, including vagueness and incompleteness of knowledge. The Mathematical Theory of Evidence or the Dempster-Shafer Theory (DST) [1, 12] has been intensely investigated in the past as a means of expressing incomplete knowledge. The interesting property in this context is that DST formally fits into the framework of graphoidal structures [13] which implies possibilities of efficient reasoning by local computations in large multivariate belief distributions given a factorization of the belief distribution into low dimensional component conditional belief functions. But the concept of conditional belief functions is generally not usable because composition of conditional belief functions is not granted to yield joint multivariate belief distribution, as some values of the belief distribution may turn out to be negative [4, 13, 15]. To overcome this problem creation of an adequate frequency model is needed. In this paper we suggest that a Dempster-Shafer distribution results from ''clustering'' (merging) of objects sharing common features. Upon ''clustering'' two (or more) objects become indistinguishable (will be counted as one) but some attributes will behave as if they have more than one value at once. The next elements of the model needed are the concept of conditional independence and that of merger conditions. It is assumed that before merger the objects move closer in such a way that conditional distributions of features for the objects to merge are identical. The traditional conditional independence of feature variables is assumed before merger (thereafter only the DST conditional independence holds). Furthermore it is necessary that the objects get ''closer'' before the merger independly for each feature variable and only those areas merge where the conditional distributions get identical in each variable. The paper demonstrates that within this model, the graphoidal properties hold and a sufficient condition for non-negativity of the graphoidally represented belief function is presented and its validity demonstrated.