The process of diagnosis involves learning about the state of a system from various observations of symptoms or findings about the system. Sophisticated Bayesian (and other) algorithms have been developed to revise and maintain beliefs about the system as observations are made. Nonetheless, diagnostic models have tended to ignore some common sense reasoning exploited by human diagnosticians; In particular, one can learn from which observations have not been made, in the spirit of conversational implicature. There are two concepts that we describe to extract information from the observations not made. First, some symptoms, if present, are more likely to be reported before others. Second, most human diagnosticians and expert systems are economical in their data-gathering, searching first where they are more likely to find symptoms present. Thus, there is a desirable bias toward reporting symptoms that are present. We develop a simple model for these concepts that can significantly improve diagnostic inference.

Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal mixtures of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.

We analyze differences between two information-theoretically motivated approaches to statistical inference and model selection: the Minimum Description Length (MDL) principle, and the Minimum Message Length (MML) principle. Based on this analysis, we present two revised versions of MML: a pointwise estimator which gives the MML-optimal single parameter model, and a volumewise estimator which gives the MML-optimal region in the parameter space. Our empirical results suggest that with small data sets, the MDL approach yields more accurate predictions than the MML estimators. The empirical results also demonstrate that the revised MML estimators introduced here perform better than the original MML estimator suggested by Wallace and Freeman.

In this paper we investigate the geometry of the likelihood of the unknown parameters in a simple class of Bayesian directed graphs with hidden variables. This enables us, before any numerical algorithms are employed, to obtain certain insights in the nature of the unidentifiability inherent in such models, the way posterior densities will be sensitive to prior densities and the typical geometrical form these posterior densities might take. Many of these insights carry over into more complicated Bayesian networks with systematic missing data.

We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined.

We study two-layer belief networks of binary random variables in which the conditional probabilities Pr[childlparents] depend monotonically on weighted sums of the parents. In large networks where exact probabilistic inference is intractable, we show how to compute upper and lower bounds on many probabilities of interest. In particular, using methods from large deviation theory, we derive rigorous bounds on marginal probabilities such as Pr[children] and prove rates of convergence for the accuracy of our bounds as a function of network size. Our results apply to networks with generic transfer function parameterizations of the conditional probability tables, such as sigmoid and noisy-OR. They also explicitly illustrate the types of averaging behavior that can simplify the problem of inference in large networks.

The efficiency of algorithms using secondary structures for probabilistic inference in Bayesian networks can be improved by exploiting independence relations induced by evidence and the direction of the links in the original network. In this paper we present an algorithm that on-line exploits independence relations induced by evidence and the direction of the links in the original network to reduce both time and space costs. Instead of multiplying the conditional probability distributions for the various cliques, we determine on-line which potentials to multiply when a message is to be produced. The performance improvement of the algorithm is emphasized through empirical evaluations involving large real world Bayesian networks, and we compare the method with the HUGIN and Shafer-Shenoy inference algorithms.

It was recently shown that the problem of decoding messages transmitted through a noisy channel can be formulated as a belief updating task over a probabilistic network [McEliece]. Moreover, it was observed that iterative application of the (linear time) Pearl's belief propagation algorithm designed for polytrees outperformed state of the art decoding algorithms, even though the corresponding networks may have many cycles. This paper demonstrates empirically that an approximation algorithm approx-mpe for solving the most probable explanation (MPE) problem, developed within the recently proposed mini-bucket elimination framework [Dechter96], outperforms iterative belief propagation on classes of coding networks that have bounded induced width. Our experiments suggest that approximate MPE decoders can be good competitors to the approximate belief updating decoders.

Qualitative probabilistic reasoning in a Bayesian network often reveals tradeoffs: relationships that are ambiguous due to competing qualitative influences. We present two techniques that combine qualitative and numeric probabilistic reasoning to resolve such tradeoffs, inferring the qualitative relationship between nodes in a Bayesian network. The first approach incrementally marginalizes nodes that contribute to the ambiguous qualitative relationships. The second approach evaluates approximate Bayesian networks for bounds of probability distributions, and uses these bounds to determinate qualitative relationships in question. This approach is also incremental in that the algorithm refines the state spaces of random variables for tighter bounds until the qualitative relationships are resolved. Both approaches provide systematic methods for tradeoff resolution at potentially lower computational cost than application of purely numeric methods.

In recent years there has been a flurry of works on learning Bayesian networks from data. One of the hard problems in this area is how to effectively learn the structure of a belief network from incomplete data- that is, in the presence of missing values or hidden variables. In a recent paper, I introduced an algorithm called Structural EM that combines the standard Expectation Maximization (EM) algorithm, which optimizes parameters, with structure search for model selection. That algorithm learns networks based on penalized likelihood scores, which include the BIC/MDL score and various approximations to the Bayesian score. In this paper, I extend Structural EM to deal directly with Bayesian model selection. I prove the convergence of the resulting algorithm and show how to apply it for learning a large class of probabilistic models, including Bayesian networks and some variants thereof.