Comprehensible explanations of probabilistic reasoning are a prerequisite for wider acceptance of Bayesian methods in expert systems and decision support systems. A study of human reasoning under uncertainty suggests two different strategies for explaining probabilistic reasoning: The first, qualitative belief propagation, traces the qualitative effect of evidence through a belief network from one variable to the next. This propagation algorithm is an alternative to the graph reduction algorithms of Wellman (1988) for inference in qualitative probabilistic networks. It is based on a qualitative analysis of intercausal reasoning, which is a generalization of Pearl's "explaining away", and an alternative to Wellman's definition of qualitative synergy. The other, Scenario-based reasoning, involves the generation of alternative causal "stories" accounting for the evidence. Comparing a few of the most probable scenarios provides an approximate way to explain the results of probabilistic reasoning. Both schemes employ causal as well as probabilistic knowledge. Probabilities may be presented as phrases and/or numbers. Users can control the style, abstraction and completeness of explanations.

Functional dependencies restrict the potential interactions among variables connected in a probabilistic network. This restriction can be exploited in qualitative probabilistic reasoning by introducing deterministic variables and modifying the inference rules to produce stronger conclusions in the presence of functional relations. I describe how to accomplish these modifications in qualitative probabilistic networks by exhibiting the update procedures for graphical transformations involving probabilistic and deterministic variables and combinations. A simple example demonstrates that the augmented scheme can reduce qualitative ambiguity that would arise without the special treatment of functional dependency. Analysis of qualitative synergy reveals that new higher-order relations are required to reason effectively about synergistic interactions among deterministic variables.

Mechanisms for the automation of uncertainty are required for expert systems. Sometimes these mechanisms need to obey the properties of probabilistic reasoning. A purely numeric mechanism, like those proposed so far, cannot provide a probabilistic logic with truth functional connectives. We propose an alternative mechanism, Incidence Calculus, which is based on a representation of uncertainty using sets of points, which might represent situations, models or possible worlds. Incidence Calculus does provide a probabilistic logic with truth functional connectives.

ABSTRACT The issue of confidence factors in Knowledge Based Systems has become increasingly important and Dempster-Shafer (DS) theory has become increasingly popular as a basis for these factors. This paper discusses the need for an empirical interpretation of any theory of confidence factors applied to Knowledge Based Systems and describes an empirical interpretation of DS theory suggesting that the theory-has been seriously misinterpreted. For the essentially syntactic DS theory, the empirical model developed is based on the semantics of sample spaces. This model is used to show that, if belief functions are based on reasonably accurate sampling or observation of a sample space, then the beliefs and upper probabilities as computed according to OS theory cannot be interpreted as frequency ratios. Since a number of proposed applications of OS theory use belief functions in situations with statistically derived evidence and seem to appeal to statistical intuition to provide an interpretation of the results, it is likely that OS theory has often been misapplied. CONFIDENCE FACTORS, EMPIRICISM AND THE DEMPSTER-SHAFER THEORY OF EVIDENCE The issue of confidence factors in Knowledge Based Systems has become increasingly important and Dempster-Shafer (DS) theory has become increasingly popular as a basis for these factors.

Expert systems applications that involve uncertain inference can be represented by a multidimensional contingency table. These tables offer a general approach to inferring with uncertain evidence, because they can embody any form of association between any number of pieces of evidence and conclusions. (Simpler models may be required, however, if the number of pieces of evidence bearing on a conclusion is large.) This paper presents a method of using these tables to make uncertain inferences without assumptions of conditional independence among pieces of evidence or heuristic combining rules. As evidence is accumulated, new joint probabilities are calculated so as to maintain any dependencies among the pieces of evidence that are found in the contingency table. The new conditional probability of the conclusion is then calculated directly from these new joint probabilities and the conditional probabilities in the contingency table.

The form and justification of inductive inference rules depend strongly on the representation of uncertainty. This paper examines one generic representation, namely, incomplete information. The notion can be formalized by presuming that the relevant probabilities in a decision problem are known only to the extent that they belong to a class K of probability distributions. The concept is a generalization of a frequent suggestion that uncertainty be represented by intervals or ranges on probabilities. To make the representation useful for decision making, an inductive rule can be formulated which determines, in a well-defined manner, a best approximation to the unknown probability, given the set K. In addition, the knowledge set notion entails a natural procedure for updating -- modifying the set K given new evidence. Several non-intuitive consequences of updating emphasize the differences between inference with complete and inference with incomplete information.

Several different uncertain inference systems (UISs) have been developed for representing uncertainty in rule-based expert systems. Some of these, such as Mycin's Certainty Factors, Prospector, and Bayes' Networks were designed as approximations to probability, and others, such as Fuzzy Set Theory and DempsterShafer Belief Functions were not. How different are these UISs in practice, and does it matter which you use? When combining and propagating uncertain information, each UIS must, at least by implication, make certain assumptions about correlations not explicily specified. The maximum entropy principle with minimum cross-entropy updating, provides a way of making assumptions about the missing specification that minimizes the additional information assumed, and thus offers a standard against which the other UISs can be compared. We describe a framework for the experimental comparison of the performance of different UISs, and provide some illustrative results.

This paper is concerned with two theories of probability judgment: the Bayesian theory and the theory of belief functions. It illustrates these theories with some simple examples and discusses some of the issues that arise when we try to implement them in expert systems. The Bayesian theory is well known; its main ideas go back to the work of Thomas Bayes (1702-1761). The theory of belief functions, often called the Dempster-Shafer theory in the artificial intelligence community, is less well known, but it has even older antecedents; belief-function arguments appear in the work of George Hooper (16401723) and James Bernoulli (1654-1705). For elementary expositions of the theory of belief functions, see Shafer (1976, 1985).

Evidential reasoning in expert systems has often used ad-hoc uncertainty calculi. Although it is generally accepted that probability theory provides a firm theoretical foundation, researchers have found some problems with its use as a workable uncertainty calculus. Among these problems are representation of ignorance, consistency of probabilistic judgements, and adjustment of a priori judgements with experience. The application of metaprobability theory to evidential reasoning is a new approach to solving these problems. Metaprobability theory can be viewed as a way to provide soft or hard constraints on beliefs in much the same manner as the Dempster-Shafer theory provides constraints on probability masses on subsets of the state space. Thus, we use the Dempster-Shafer theory, an alternative theory of evidential reasoning to illuminate metaprobability theory as a theory of evidential reasoning. The goal of this paper is to compare how metaprobability theory and Dempster-Shafer theory handle the adjustment of beliefs with evidence with respect to a particular thought experiment. Sections 2 and 3 give brief descriptions of the metaprobability and Dempster-Shafer theories. Metaprobability theory deals with higher order probabilities applied to evidential reasoning. Dempster-Shafer theory is a generalization of probability theory which has evolved from a theory of upper and lower probabilities. Section 4 describes a thought experiment and the metaprobability and DempsterShafer analysis of the experiment. The thought experiment focuses on forming beliefs about a population with 6 types of members {1, 2, 3, 4, 5, 6}. A type is uniquely defined by the values of three features: A, B, C. That is, if the three features of one member of the population were known then its type could be ascertained. Each of the three features has two possible values, (e.g. A can be either "a0" or "al"). Beliefs are formed from evidence accrued from two sensors: sensor A, and sensor B. Each sensor senses the corresponding defining feature. Sensor A reports that half of its observations are "a0" and half the observations are 'al'. Sensor B reports that half of its observations are ``b0,' and half are "bl". Based on these two pieces of evidence, what should be the beliefs on the distribution of types in the population? Note that the third feature is not observed by any sensor.

A FRAMEWORK FOR NONMONOTONIC REASONING ABOUT PROBABILISTIC ASSUMPTIONS-- The Problem Marvin S. Cohen Decision Science Consortium, Inc. Attempts to replicate probabilistic reasoning in expert systems have typically overlooked a critical ingredient of that process. The application of such models is often an iterative process, in which the plausibility of the results confirms or disconfirms the validity of assumptions made in building the model. In current expert systems, by contrast, probabilistic information is encapsulated within modular rules (involving, for example, "certainty factors"), and there is no mechanism for reviewing the overall form of the probability argument or the validity of the judgments entering into it. It involves the design of an expert system inference framework in which probabilistic statements and rules are regarded as assumptions which are explicitly tracked and reevaluated when they lead to conflict among different sources of evidence or lines of reasoning. Two conceptions of conflict and conflict resolution have been implicit in most approaches to this area. From one point of view, divergence among lines of reasoning can be regarded as stochastic; it is expected to occur some small percentage of the time, due to the chance accumulation of small errors or "noise" in an imperfect process of "measurement". From another point of view, however, divergence can be regarded as a result of faulty beliefs; that is, conflicting results are taken as evidence that one or more premises or forms of argument that led to the conflict are mistaken.