Theory refinement is the task of updating a domain theory in the light of new cases, to be done automatically or with some expert assistance. The problem of theory refinement under uncertainty is reviewed here in the context of Bayesian statistics, a theory of belief revision. The problem is reduced to an incremental learning task as follows: the learning system is initially primed with a partial theory supplied by a domain expert, and thereafter maintains its own internal representation of alternative theories which is able to be interrogated by the domain expert and able to be incrementally refined from data. Algorithms for refinement of Bayesian networks are presented to illustrate what is meant by "partial theory", "alternative theory representation", etc. The algorithms are an incremental variant of batch learning algorithms from the literature so can work well in batch and incremental mode.

Research on Symbolic Probabilistic Inference (SPI) [2, 3] has provided an algorithm for 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. Unlike traditional Bayesian network inferencing algorithms, SPI algorithm is goal directed, performing only those calculations that are required to respond to queries. Research to date on SPI applies to Bayesian networks with discrete-valued variables and does not address variables with continuous values. In this papers, we extend the SPI algorithm to handle Bayesian networks made up of continuous variables where the relationships between the variables are restricted to be ?linear gaussian?. We call this variation of the SPI algorithm, SPI Continuous (SPIC). SPIC modifies the three basic SPI operations: multiplication, summation, and substitution. However, SPIC retains the framework of the SPI algorithm, namely building the search tree and recursive query mechanism and therefore retains the goal-directed and incrementality features of SPI.

In probabilistic logic entailments, even moderate size problems can yield linear constraint systems with so many variables that exact methods are impractical. This difficulty can be remedied in many cases of interest by introducing a three valued logic (true, false, and "don't care"). The three-valued approach allows the construction of "compressed" constraint systems which have the same solution sets as their two-valued counterparts, but which may involve dramatically fewer variables. Techniques to calculate point estimates for the posterior probabilities of entailed sentences are discussed.

This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature of plausible reasoning. These issues have a common underlying theme: there should be an underlying belief calculus of which the many different reasoning forms are special cases, sometimes approximate. The system presented allows reasoning about defaults, likelihood, necessity and possibility in a manner similar to the earlier work of Adams. The system is based on the belief calculus of subjective Bayesian probability which itself is based on a few simple assumptions about how belief should be manipulated. Approximations, semantics, consistency and consequence results are presented for the system. While this puts these often discussed plausible reasoning forms on a probabilistic footing, useful application to practical problems remains an issue.

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.

An approach to reasoning with default rules where the proportion of exceptions, or more generally the probability of encountering an exception, can be at least roughly assessed is presented. It is based on local uncertainty propagation rules which provide the best bracketing of a conditional probability of interest from the knowledge of the bracketing of some other conditional probabilities. A procedure that uses two such propagation rules repeatedly is proposed in order to estimate any simple conditional probability of interest from the available knowledge. The iterative procedure, that does not require independence assumptions, looks promising with respect to the linear programming method. Improved bounds for conditional probabilities are given when independence assumptions hold.

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.

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.

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.

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.