Many writers have observed that default logics appear to contain the "lottery paradox" of probability theory. This arises when a default "proof by contradiction" lets us conclude that a typical X is not a Y where Y is an unusual subclass of X. We show that there is a similar problem with default "proof by cases" and construct a setting where we might draw a different conclusion knowing a disjunction than we would knowing any particular disjunct. Though Reiter's original formalism is capable of representing this distinction, other approaches are not. To represent and reason about this case, default logicians must specify how a "typical" individual is selected. The problem is closely related to Simpson's paradox of probability theory. If we accept a simple probabilistic account of defaults based on the notion that one proposition may favour or increase belief in another, the "multiple extension problem" for both conjunctive and disjunctive knowledge vanishes.

We derive axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.

The role of inferencing with uncertainty is becoming more important in rule-based expert systems (ES), since knowledge given by a human expert is often uncertain or imprecise. We have succeeded in designing a VLSI chip which can perform an entire inference process based on fuzzy logic. The design of the VLSI fuzzy inference engine emphasizes simplicity, extensibility, and efficiency (operational speed and layout area). It is fabricated in 2.5 um CMOS technology. The inference engine consists of three major components; a rule set memory, an inference processor, and a controller. In this implementation, a rule set memory is realized by a read only memory (ROM). The controller consists of two counters. In the inference processor, one data path is laid out for each rule. The number of the inference rule can be increased adding more data paths to the inference processor. All rules are executed in parallel, but each rule is processed serially. The logical structure of fuzzy inference proposed in the current paper maps nicely onto the VLSI structure. A two-phase nonoverlapping clocking scheme is used. Timing tests indicate that the inference engine can operate at approximately 20.8 MHz. This translates to an execution speed of approximately 80,000 Fuzzy Logical Inferences Per Second (FLIPS), and indicates that the inference engine is suitable for a demanding real-time application. The potential applications include decision-making in the area of command and control for intelligent robot systems, process control, missile and aircraft guidance, and other high performance machines.

We describe a representation and a set of inference methods that combine logic programming techniques with probabilistic network representations for uncertainty (influence diagrams). The techniques emphasize the dynamic construction and solution of probabilistic and decision-theoretic models for complex and uncertain domains. Given a query, a logical proof is produced if possible; if not, an influence diagram based on the query and the knowledge of the decision domain is produced and subsequently solved. A uniform declarative, first-order, knowledge representation is combined with a set of integrated inference procedures for logical, probabilistic, and decision-theoretic reasoning.

Reasoning under uncertainty in Al hats come to mean assessing the credibility of hypotheses inferred from evidence. But techniques for assessing credibility do not tell a problem solver what to do when it is uncertain. This is the focus of our current research. We have developed a medical expert system called MUM, for Managing Uncertainty in Medicine, that plans diagnostic sequences of questions, tests, and treatments. This paper describes the kinds of problems that MUM was designed to solve and gives a brief description of its architecture. More recently, we have built an empty version of MUM called MU, and used it to reimplement MUM and a small diagnostic system for plant pathology. The latter part of the paper describes the features of MU that make it appropriate for building expert systems that manage uncertainty.

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.

This paper addresses fundamental issues on the nature of the concepts and structures of fuzzy logic, focusing, in particular, on the conceptual and functional differences that exist between probabilistic and possibilistic approaches. A semantic model provides the basic framework to define possibilistic structures and concepts by means of a function that quantifies proximity, closeness, or resemblance between pairs of possible worlds. The resulting model is a natural extension, based on multiple conceivability relations, of the modal logic concepts of necessity and possibility. By contrast, chance-oriented probabilistic concepts and structures rely on measures of set extension that quantify the proportion of possible worlds where a proposition is true. Resemblance between possible worlds is quantified by a generalized similarity relation: a function that assigns a number between O and 1 to every pair of possible worlds. Using this similarity relation, which is a form of numerical complement of a classic metric or distance, it is possible to define and interpret the major constructs and methods of fuzzy logic: conditional and unconditioned possibility and necessity distributions and the generalized modus ponens of Zadeh.

We analyzed the convergence properties of likelihood- weighting algorithms on a two-level, multiply connected, belief-network representation of the QMR knowledge base of internal medicine. Specifically, on two difficult diagnostic cases, we examined the effects of Markov blanket scoring, importance sampling, demonstrating that the Markov blanket scoring and self-importance sampling significantly improve the convergence of the simulation on our model.

This paper presents some ideas and results of using uncertainty management methods in the presence of data in preference to other statistical and machine learning methods. A medical domain is used as a test-bed with data available from a large hospital database system which collects symptom and outcome information about patients. Data is often missing, of many variable types and sample sizes for particular outcomes is not large. Uncertainty management methods are useful for such domains and have the added advantage of allowing for expert modification of belief values originally obtained from data. Methodological considerations for using belief functions on statistical data are dealt with in some detail. Expert opinions are Incorporated at various levels of the project development and results are reported on an application to liver disease diagnosis. Recent results contrasting the use of weights of evidence and logistic regression on another medical domain are also presented.

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.