We describe a method for incrementally constructing belief networks. We have developed a network-construction language similar to a forward-chaining language using data dependencies, but with additional features for specifying distributions. Using this language, we can define parameterized classes of probabilistic models. These parameterized models make it possible to apply probabilistic reasoning to problems for which it is impractical to have a single large static model.

After a brief introduction to causal probabilistic networks and the HUGIN approach, the problem of conflicting data is discussed. A measure of conflict is defined, and it is used in the medical diagnostic system MUNIN. Finally, it is discussed how to distinguish between conflicting data and a rare case.

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.

Our previous work on classifying complex ship images [1,2] has evolved into an effort to develop software tools for building and solving generic classification problems. Managing the uncertainty associated with feature data and other evidence is an important issue in this endeavor. Bayesian techniques for managing uncertainty [7,12,13] have proven to be useful for managing several of the belief maintenance requirements of classification problem solving. One such requirement is the need to give qualitative explanations of what is believed. Pearl [11] addresses this need by computing what he calls a belief commitment-the most probable instantiation of all hypothesis variables given the evidence available. Before belief commitments can be computed, the straightforward implementation of Pearl's procedure involves finding an analytical solution to some often difficult optimization problems. We describe an efficient implementation of this procedure using tensor products that solves these problems enumeratively and avoids the need for case by case analysis. The procedure is thereby made more practical to use in the general case.

Stochastic simulation approaches perform probabilistic inference in Bayesian networks by estimating the probability of an event based on the frequency that the event occurs in a set of simulation trials. This paper describes the evidence weighting mechanism, for augmenting the logic sampling stochastic simulation algorithm [Henrion, 1986]. Evidence weighting modifies the logic sampling algorithm by weighting each simulation trial by the likelihood of a network's evidence given the sampled state node values for that trial. We also describe an enhancement to the basic algorithm which uses the evidential integration technique [Chin and Cooper, 1987]. A comparison of the basic evidence weighting mechanism with the Markov blanket algorithm [Pearl, 1987], the logic sampling algorithm, and the evidence integration algorithm is presented. The comparison is aided by analyzing the performance of the algorithms in a simple example network.

Much of the controversy about methods for automated decision making has focused on specific calculi for combining beliefs or propagating uncertainty. We broaden the debate by (1) exploring the constellation of secondary tasks surrounding any primary decision problem, and (2) identifying knowledge engineering concerns that present additional representational tradeoffs. We argue on pragmatic grounds that the attempt to support all of these tasks within a single calculus is misguided. In the process, we note several uncertain reasoning objectives that conflict with the Bayesian ideal of complete specification of probabilities and utilities. In response, we advocate treating the uncertainty calculus as an object language for reasoning mechanisms that support the secondary tasks. Arguments against Bayesian decision theory are weakened when the calculus is relegated to this role. Architectures for uncertainty handling that take statements in the calculus as objects to be reasoned about offer the prospect of retaining normative status with respect to decision making while supporting the other tasks in uncertain reasoning.

A number of algorithms have been developed to solve probabilistic inference problems on belief networks. These algorithms can be divided into two main groups: exact techniques which exploit the conditional independence revealed when the graph structure is relatively sparse, and probabilistic sampling techniques which exploit the "conductance" of an embedded Markov chain when the conditional probabilities have non-extreme values. In this paper, we investigate a family of "forward" Monte Carlo sampling techniques similar to Logic Sampling [Henrion, 1988] which appear to perform well even in some multiply connected networks with extreme conditional probabilities, and thus would be generally applicable. We consider several enhancements which reduce the posterior variance using this approach and propose a framework and criteria for choosing when to use those enhancements.

In [12], Nilsson proposed the probabilistic logic in which the truth values of logical propositions are probability values between 0 and 1. It is applicable to any logical system for which the consistency of a finite set of propositions can be established. The probabilistic inference scheme reduces to the ordinary logical inference when the probabilities of all propositions are either 0 or 1. This logic has the same limitations of other probabilistic reasoning systems of the Bayesian approach. For common sense reasoning, consistency is not a very natural assumption. We have some well known examples: {Dick is a Quaker, Quakers are pacifists, Republicans are not pacifists, Dick is a Republican}and {Tweety is a bird, birds can fly, Tweety is a penguin}. In this paper, we shall propose some extensions of the probabilistic logic. In the second section, we shall consider the space of all interpretations, consistent or not. In terms of frames of discernment, the basic probability assignment (bpa) and belief function can be defined. Dempster's combination rule is applicable. This extension of probabilistic logic is called the evidential logic in [ 1]. For each proposition s, its belief function is represented by an interval [Spt(s), Pls(s)]. When all such intervals collapse to single points, the evidential logic reduces to probabilistic logic (in the generalized version of not necessarily consistent interpretations). Certainly, we get Nilsson's probabilistic logic by further restricting to consistent interpretations. In the third section, we shall give a probabilistic interpretation of probabilistic logic in terms of multi-dimensional random variables. This interpretation brings the probabilistic logic into the framework of probability theory. Let us consider a finite set S = {sl, s2, ..., Sn) of logical propositions. Each proposition may have true or false values; and may be considered as a random variable. We have a probability distribution for each proposition. The e-dimensional random variable (sl,..., Sn) may take values in the space of all interpretations of 2n binary vectors. We may compute absolute (marginal), conditional and joint probability distributions. It turns out that the permissible probabilistic interpretation vector of Nilsson [12] consists of the joint probabilities of S. Inconsistent interpretations will not appear, by setting their joint probabilities to be zeros. By summing appropriate joint probabilities, we get probabilities of individual propositions or subsets of propositions. Since the Bayes formula and other techniques are valid for e-dimensional random variables, the probabilistic logic is actually very close to the Bayesian inference schemes. In the last section, we shall consider a relaxation scheme for probabilistic logic. In this system, not only new evidences will update the belief measures of a collection of propositions, but also constraint satisfaction among these propositions in the relational network will revise these measures. This mechanism is similar to human reasoning which is an evaluative process converging to the most satisfactory result. The main idea arises from the consistent labeling problem in computer vision. This method is originally applied to scene analysis of line drawings. Later, it is applied to matching, constraint satisfaction and multi sensor fusion by several authors [8], [16] (and see references cited there). Recently, this method is used in knowledge aggregation by Landy and Hummel [9].

We propose an abductive diagnosis theory that integrates probabilistic, causal and taxonomic knowledge. Probabilistic knowledge allows us to select the most likely explanation; causal knowledge allows us to make reasonable independence assumptions; taxonomic knowledge allows causation to be modeled at different levels of detail, and allows observations be described in different levels of precision. Unlike most other approaches where a causal explanation is a hypothesis that one or more causative events occurred, we define an explanation of a set of observations to be an occurrence of a chain of causation events. These causation events constitute a scenario where all the observations are true. We show that the probabilities of the scenarios can be computed from the conditional probabilities of the causation events. Abductive reasoning is inherently complex even if only modest expressive power is allowed. However, our abduction algorithm is exponential only in the number of observations to be explained, and is polynomial in the size of the knowledge base. This contrasts with many other abduction procedures that are exponential in the size of the knowledge base.

We introduce a graceful approach to probabilistic inference called bounded conditioning. Bounded conditioning monotonically refines the bounds on posterior probabilities in a belief network with computation, and converges on final probabilities of interest with the allocation of a complete resource fraction. The approach allows a reasoner to exchange arbitrary quantities of computational resource for incremental gains in inference quality. As such, bounded conditioning holds promise as a useful inference technique for reasoning under the general conditions of uncertain and varying reasoning resources. The algorithm solves a probabilistic bounding problem in complex belief networks by breaking the problem into a set of mutually exclusive, tractable subproblems and ordering their solution by the expected effect that each subproblem will have on the final answer. We introduce the algorithm, discuss its characterization, and present its performance on several belief networks, including a complex model for reasoning about problems in intensive-care medicine.