In this paper the elicitation of probabilities from human experts is considered as a measurement process, which may be disturbed by random 'measurement noise'. Using Bayesian concepts a second order probability distribution is derived reflecting the uncertainty of the input probabilities. The algorithm is based on an approximate sample representation of the basic probabilities. This sample is continuously modified by a stochastic simulation procedure, the Metropolis algorithm, such that the sequence of successive samples corresponds to the desired posterior distribution. The procedure is able to combine inconsistent probabilities according to their reliability and is applicable to general inference networks with arbitrary structure. Dempster-Shafer probability mass functions may be included using specific measurement distributions. The properties of the approach are demonstrated by numerical experiments.

A framework is presented for a computational theory of probabilistic argument. The Probabilistic Reasoning Environment encodes knowledge at three levels. At the deepest level are a set of schemata encoding the system's domain knowledge. This knowledge is used to build a set of second-level arguments, which are structured for efficient recapture of the knowledge used to construct them. Finally, at the top level is a Bayesian network constructed from the arguments. The system is designed to facilitate not just propagation of beliefs and assimilation of evidence, but also the dynamic process of constructing a belief network, evaluating its adequacy, and revising it when necessary.

Incidence Calculus and Dempster-Shafer Theory of Evidence are both theories to describe agents' degrees of belief in propositions, thus being appropriate to represent uncertainty in reasoning systems. This paper presents a straightforward equivalence proof between some special cases of these theories.

In this paper, we suggest marrying Dempster-Shafer (DS) theory with Knowledge Representation (KR). Born out of this marriage is the definition of "Dempster-Shafer Belief Bases", abstract data types representing uncertain knowledge that use DS theory for representing strength of belief about our knowledge, and the linguistic structures of an arbitrary KR system for representing the knowledge itself. A formal result guarantees that both the properties of the given KR system and of DS theory are preserved. The general model is exemplified by defining DS Belief Bases where First Order Logic and (an extension of) KRYPTON are used as KR systems. The implementation problem is also touched upon.

In this paper we associate with every (directed) graph G a transformation called the Mobius transformation of the graph G. The Mobius transformation of the graph (O) is of major significance for Dempster-Shafer theory of evidence. However, because it is computationally very heavy, the Mobius transformation together with Dempster's rule of combination is a major obstacle to the use of Dempster-Shafer theory for handling uncertainty in expert systems. The major contribution of this paper is the discovery of the 'fast Mobius transformations' of (O). These 'fast Mobius transformations' are the fastest algorithms for computing the Mobius transformation of (O). As an easy but useful application, we provide, via the commonality function, an algorithm for computing Dempster's rule of combination which is much faster than the usual one.

Dempster-Shafer's model aims at quantifying degrees of belief But there are so many interpretations of Dempster-Shafer's theory in the literature that it seems useful to present the various contenders in order to clarify their respective positions. We shall successively consider the classical probability model, the upper and lower probabilities model, Dempster's model, the transferable belief model, the evidentiary value model, the provability or necessity model. None of these models has received the qualification of Dempster-Shafer. In fact the transferable belief model is our interpretation not of Dempster's work but of Shafer's work as presented in his book (Shafer 1976, Smets 1988). It is a ?purified' form of Dempster-Shafer's model in which any connection with probability concept has been deleted. Any model for belief has at least two components: one static that describes our state of belief, the other dynamic that explains how to update our belief given new pieces of information. We insist on the fact that both components must be considered in order to study these models. Too many authors restrict themselves to the static component and conclude that Dempster-Shafer theory is the same as some other theory. But once the dynamic component is considered, these conclusions break down. Any comparison based only on the static component is too restricted. The dynamic component must also be considered as the originality of the models based on belief functions lies in its dynamic component.

We define a new notion of conditional belief, which plays the same role for Dempster-Shafer belief functions as conditional probability does for probability functions. Our definition is different from the standard definition given by Dempster, and avoids many of the well-known problems of that definition. Just as the conditional probability Pr (lB) is a probability function which is the result of conditioning on B being true, so too our conditional belief function Bel (lB) is a belief function which is the result of conditioning on B being true. We define the conditional belief as the lower envelope (that is, the inf) of a family of conditional probability functions, and provide a closed form expression for it. An alternate way of understanding our definition of conditional belief is provided by considering ideas from an earlier paper [Fagin and Halpern, 1989], where we connect belief functions with inner measures. In particular, we show here how to extend the definition of conditional probability to non measurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given our lack of information. Our definition of conditional belief turns out to be an exact analogue of our definition of inner conditional probability.

A method of calculating probability values from a system of marginal constraints is presented. Previous systems for finding the probability of a single attribute have either made an independence assumption concerning the evidence or have required, in the worst case, time exponential in the number of attributes of the system. In this paper a closed form solution to the probability of an attribute given the evidence is found. The closed form solution, however does not enforce the (non-linear) constraint that all terms in the underlying distribution be positive. The equation requires O(r^3) steps to evaluate, where r is the number of independent marginal constraints describing the system at the time of evaluation. Furthermore, a marginal constraint may be exchanged with a new constraint, and a new solution calculated in O(r^2) steps. This method is appropriate for calculating probabilities in a real time expert system

When expert systems based on causal probabilistic networks (CPNs) reach a certain size and complexity, the "combinatorial explosion monster" tends to be present. We propose an approximation scheme that identifies rarely occurring cases and excludes these from being processed as ordinary cases in a CPN-based expert system. Depending on the topology and the probability distributions of the CPN, the numbers (representing probabilities of state combinations) in the underlying numerical representation can become very small. Annihilating these numbers and utilizing the resulting sparseness through data structuring techniques often results in several orders of magnitude of improvement in the consumption of computer resources. Bounds on the errors introduced into a CPN-based expert system through approximations are established. Finally, reports on empirical studies of applying the approximation scheme to a real-world CPN are given.

Kutato is a system that takes as input a database of cases and produces a belief network that captures many of the dependence relations represented by those data. This system incorporates a module for determining the entropy of a belief network and a module for constructing belief networks based on entropy calculations. Kutato constructs an initial belief network in which all variables in the database are assumed to be marginally independent. The entropy of this belief network is calculated, and that arc is added that minimizes the entropy of the resulting belief network. Conditional probabilities for an arc are obtained directly from the database. This process continues until an entropy-based threshold is reached. We have tested the system by generating databases from networks using the probabilistic logic-sampling method, and then using those databases as input to Kutato. The system consistently reproduces the original belief networks with high fidelity.