The clique tree algorithm is the standard method for doing inference in Bayesian networks. It works by manipulating clique potentials — distributions over the variables in a clique. While this approach works well for many networks, it is limited by the need to maintain an exact representation of the clique potentials. This paper presents a new unified approach that combines approximate inference and the clique tree algorithm, thereby circumventing this limitation. Many known approximate inference algorithms can be viewed as instances of this approach. The algorithm essentially does clique tree propagation, using approximate inference to estimate the densities in each clique. In many settings, the computation of the approximate clique potential can be done easily using statistical importance sampling. Iterations are used to gradually improve the quality of the estimation. 1

We show how to use a variational approximation to the logistic function to perform approximate inference in Bayesian networks containing discrete nodes with continuous parents. Essentially, we convert the logistic function to a Gaussian, which facilitates exact inference, and then iteratively adjust the variational parameters to improve the quality of the approximation. We demonstrate experimentally that this approximation is much faster than sampling, but comparable in accuracy. We also introduce a simple new technique for handling evidence, which allows us to handle arbitrary distributionson observed nodes, as well as achieving a significant speedup in networks with discrete variables of large cardinality. 1

Many real-world systems are naturally modeled as hybrid stochastic processes, i.e., stochastic processes that contain both discrete and continuous variables. Examples include speech recognition, target tracking, and monitoring of physical systems. The task is usually to perform probabilistic inference, i.e., infer the hidden state of the system given some noisy observations. For example, we can ask what is the probability that a certain word was pronounced given the readings of our microphone, what is the probability that a submarine is trying to surface given our sonar data, and what is the probability of a valve being open given our pressure and flow readings. Bayesian networks are

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of

Many real life domains contain a mixture of discrete and continuous variables and can be modeled as hybrid Bayesian Networks (BNs). An important subclass of hybrid BNs are conditional linear Gaussian (CLG) networks, where the conditional distribution of the continuous variables given an assignment to the discrete variables is a multivariate Gaussian. Lauritzen’s extension to the clique tree algorithm can be used for exact inference in CLG networks. However, many domains include discrete variables that depend on continuous ones, and CLG networks do not allow such dependencies to be represented. In this paper, we propose the first “exact ” inference algorithm for augmented CLG networks — CLG networks augmented by allowing discrete children of continuous parents. Our algorithm is based on Lauritzen’s algorithm, and is exact in a similar sense: it computes the exact distributions over the discrete nodes, and the exact first and second moments of the continuous ones, up to inaccuracies resulting from numerical integration used within the algorithm. In the special case of softmax CPDs, we show that integration can often be done efficiently, and that using the first two moments leads to a particularly accurate approximation. We show empirically that our algorithm achieves substantially higher accuracy at lower cost than previous algorithms for this task. 1

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Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the Shenoy-Shafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate standard PDF’s and applications of these potentials for solving inference problems in hybrid Bayesian networks. These approximations will extend the types of inference problems that can be modeled with Bayesian networks, as demonstrated using three examples.

This paper presents non-random algorithms for approximate computation in Bayesian networks. They are based on the use of probability trees to represent probability potentials, using the Kullback-Leibler cross entropy as a measure of the error of the approximation. Different alternatives are presented and tested in several experiments with difficult propagation problems. The results show how it is possible to find good approximations in short time compared with Hugin algorithm. � 2000 John Wiley & Sons, Inc. 1.

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function can be approximated with an MTE potential, which can always by marginalized in closed form. This allows propagation to be done exactly using the Shenoy-Shafer architecture for computing marginals, with no restrictions on the construction of a join tree.

norman,martin,william