This paper presents a simple framework for Horn clause abduction, with probabilities associated with hypotheses. It is shown how this representation can represent any probabilistic knowledge representable in a Bayesian belief network. The main contributions are in finding a relationship between logical and probabilistic notions of evidential reasoning. This can be used as a basis for a new way to implement Bayesian Networks that allows for approximations to the value of the posterior probabilities, and also points to a way that Bayesian networks can be extended beyond a propositional language.

The EMalgorithm is a general procedure to get maximum likelihood estimates if part of the observations on the variables of a network are missing. In this paper a stochastic version of the algorithm is adapted to probabilistic neural networks describing the associative dependency of variables. These networks have a probability distribution, which is a special case of the distribution generated by probabilistic inference networks. Hence both types of networks can be combined allowing to integrate probabilistic rules as well as unspecified associations in a sound way. The resulting network may have a number of interesting features including cycles of probabilistic rules, hidden'unobservable' variables, and uncertain and contradictory evidence. IN TRODUCTION Probabilistic inference networks (Pearl 1988) have been used to model uncertain causal relations between variables, for instance in a diagnostic system.

At last year?s Uncertainty in AI Conference, we reported the results of a sensitivity analysis study of Pathfinder. Our findings were quite unexpected-slight variations to Pathfinder?s parameters appeared to lead to substantial degradations in system performance. A careful look at our first analysis, together with the valuable feedback provided by the participants of last year?s conference, led us to conduct a follow-up study. Our follow-up differs from our initial study in two ways: (i) the probabilities 0.0 and 1.0 remained unchanged, and (ii) the variations to the probabilities that are close to both ends (0.0 or 1.0) were less than the ones close to the middle (0.5). The results of the follow-up study look more reasonable-slight variations to Pathfinder?s parameters now have little effect on its performance. Taken together, these two sets of results suggest a viable extension of a common decision analytic sensitivity analysis to the larger, more complex settings generally encountered in artificial intelligence.

The belief network is a well-known graphical structure for representing independences in a joint probability distribution. The methods, which perform probabilistic inference in belief networks, often treat the conditional probabilities which are stored in the network as certain values. However, if one takes either a subjectivistic or a limiting frequency approach to probability, one can never be certain of probability values. An algorithm should not only be capable of reporting the probabilities of the alternatives of remaining nodes when other nodes are instantiated; it should also be capable of reporting the uncertainty in these probabilities relative to the uncertainty in the probabilities which are stored in the network. In this paper a method for determining the variances in inferred probabilities is obtained under the assumption that a posterior distribution on the uncertainty variables can be approximated by the prior distribution. It is shown that this assumption is plausible if their is a reasonable amount of confidence in the probabilities which are stored in the network. Furthermore in this paper, a surprising upper bound for the prior variances in the probabilities of the alternatives of all nodes is obtained in the case where the probability distributions of the probabilities of the alternatives are beta distributions. It is shown that the prior variance in the probability at an alternative of a node is bounded above by the largest variance in an element of the conditional probability distribution for that node.

A series of monte carlo studies were performed to compare the behavior of some alternative procedures for reasoning under uncertainty. The behavior of several Bayesian, linear model and default reasoning procedures were examined in the context of increasing levels of calibration error. The most interesting result is that Bayesian procedures tended to output more extreme posterior belief values (posterior beliefs near 0.0 or 1.0) than other techniques, but the linear models were relatively less likely to output strong support for an erroneous conclusion. Also, accounting for the probabilistic dependencies between evidence items was important for both Bayesian and linear updating procedures.

We motivate and describe a theory of belief in this paper. This theory is developed with the following view of human belief in mind. Consider the belief that an event E will occur (or has occurred or is occurring). An agent either entertains this belief or does not entertain this belief (i.e., there is no "grade" in entertaining the belief). If the agent chooses to exercise "the will to believe" and entertain this belief, he/she/it is entitled to a degree of confidence c (1 > c > 0) in doing so. Adopting this view of human belief, we conjecture that whenever an agent entertains the belief that E will occur with c degree of confidence, the agent will be surprised (to the extent c) upon realizing that E did not occur.

Since exact probabilistic inference is intractable in general for large multiply connected belief nets, approximate methods are required. A promising approach is to use heuristic search among hypotheses (instantiations of the network) to find the most probable ones, as in the TopN algorithm. Search is based on the relative probabilities of hypotheses which are efficient to compute. Given upper and lower bounds on the relative probability of partial hypotheses, it is possible to obtain bounds on the absolute probabilities of hypotheses. Best-first search aimed at reducing the maximum error progressively narrows the bounds as more hypotheses are examined. Here, qualitative probabilistic analysis is employed to obtain bounds on the relative probability of partial hypotheses for the BN20 class of networks networks and a generalization replacing the noisy OR assumption by negative synergy. The approach is illustrated by application to a very large belief network, QMR-BN, which is a reformulation of the Internist-1 system for diagnosis in internal medicine.

We present a generalization of the local expression language used in the Symbolic Probabilistic Inference (SPI) approach to inference in belief nets [1l, [8]. The local expression language in SPI is the language in which the dependence of a node on its antecedents is described. The original language represented the dependence as a single monolithic conditional probability distribution. The extended language provides a set of operators (*, +, and -) which can be used to specify methods for combining partial conditional distributions. As one instance of the utility of this extension, we show how this extended language can be used to capture the semantics, representational advantages, and inferential complexity advantages of the "noisy or" relationship.

This paper presents a Bayesian method for constructing Bayesian belief networks from a database of cases. Potential applications include computer-assisted hypothesis testing, automated scientific discovery, and automated construction of probabilistic expert systems. Results are presented of a preliminary evaluation of an algorithm for constructing a belief network from a database of cases. We relate the methods in this paper to previous work, and we discuss open problems.

Recent research on the Symbolic Probabilistic Inference (SPI) algorithm[2] has focused attention on the importance of resolving general queries in Bayesian networks. SPI applies the concept of dependency-directed backward search to probabilistic inference, and is incremental with respect to both queries and observations. In response to this research we have extended the evidence potential algorithm [3] with the same features. We call the extension symbolic evidence potential inference (SEPI). SEPI like SPI can handle generic queries and is incremental with respect to queries and observations. While in SPI, operations are done on a search tree constructed from the nodes of the original network, in SEPI, a clique-tree structure obtained from the evidence potential algorithm [3] is the basic framework for recursive query processing. In this paper, we describe the systematic query and caching procedure of SEPI. SEPI begins with finding a clique tree from a Bayesian network-the standard procedure of the evidence potential algorithm. With the clique tree, various probability distributions are computed and stored in each clique. This is the ?pre-processing? step of SEPI. Once this step is done, the query can then be computed. To process a query, a recursive process similar to the SPI algorithm is used. The queries are directed to the root clique and decomposed into queries for the clique's subtrees until a particular query can be answered at the clique at which it is directed. The algorithm and the computation are simple. The SEPI algorithm will be presented in this paper along with several examples.