Nilsson's Probabilistic Entailment Extended to Dempster-Shafer Theory

Abstract

Probabilistic logic has been discussed in a recent paper by N. Nilsson [12]. An entailment scheme is proposed which can predict the probability of an event when the probabilities of certain other connected events are known. This scheme involves the use of a maximum entropy method proposed by P. Cheeseman in [3]. The model uses vectors which represent certain possible states of the world. Only such vectors are entered into the probability scheme. As a result, entailment does not always yield an acceptable result and cannot be applied to real situations which could arise. This paper investigates a technique to overcome this problem, which involves extending the idea of probabilistic logic and the maximum entropy approach to Dempster-Shaffer theory. A new entailment scheme for belief functions is used which produces well-defined results, even when only consistent worlds are being considered. The paper also reconsiders an earlier attempt by the author [6,7] to model default reasoning (and subsequent nonmonotonicity) by adding inconsistent vectors to Nilsson's model. In the extended setting, more sensible entailment values are obtained than in the previous work.