In this paper we present a method of computing the posterior probability of conditional independence of two or more continuous variables from data, examined at several resolutions. Our approach is motivated by the observation that the appearance of continuous data varies widely at various resolutions, producing very different independence estimates between the variables involved. Therefore, it is difficult to ascertain independence without examining data at several carefully selected resolutions. In our paper, we accomplish this using the exact computation of the posterior probability of independence, calculated analytically given a resolution. At each examined resolution, we assume a multinomial distribution with Dirichlet priors for the discretized table parameters, and compute the posterior using Bayesian integration. Across resolutions, we use a search procedure to approximate the Bayesian integral of probability over an exponential number of possible histograms. Our method generalizes to an arbitrary number variables in a straightforward manner. The test is suitable for Bayesian network learning algorithms that use independence tests to infer the network structure, in domains that contain any mix of continuous, ordinal and categorical variables. 1

Recently developed techniques have made it possible to quickly learn accurate probability density functions from data in low-dimensional continuous spaces. In particular, mixtures of Gaussians can be fitted to data very quickly using an accelerated EM algorithm that employs multiresolution kd-trees (Moore, 1999). In this paper, we propose a kind of Bayesian network in which low-dimensional mixtures of Gaussians over different subsets of the domain’s variables are combined into a coherent joint probability model over the entire domain. The network is also capable of modeling complex dependencies between discrete variables and continuous variables without requiring discretization of the continuous variables. We present efficient heuristic algorithms for automatically learning these networks from data, and perform comparative experiments illustrating how well these networks model real scientific data and synthetic data. We also briefly discuss some possible improvements to the networks, as well as possible applications.

my family-- especially my father, Donald. iv Abstract Many important data analysis tasks can be addressed by formulating them as probability estimation problems. For example, a popular general approach to automatic classification problems is to learn a probabilistic model of each class from data in which the classes are known, and then use Bayes's rule with these models to predict the correct classes of other data for which they are not known. Anomaly detection and scientific discovery tasks can often be addressed by learning probability models over possible events and then looking for events to which these models assign low probabilities. Many data compression algorithms such as Huffman coding and arithmetic coding rely on probabilistic models of the data stream in order achieve high compression rates.

This paper proposes a general formalism for representation, inference and learning with general hybrid Bayesian networks in which continuous and discrete variables may appear anywhere in a directed acyclic graph. The formalism fuzzies a hybrid Bayesian network into two alternative forms: The rst form replaces each continuous variable in the given directed acyclic graph (DAG) by a partner discrete variable and adding a directed link from the partner discrete variable to the continuous one. The mapping between two variables is not crisp quantization but is approximated (fuzzied) by a conditional Gaussian (CG) distribution. The CG model is equivalent to a fuzzy set but no fuzzy logic formalism is employed. The conditional distribution of a discrete variable given its discrete parents is still assumed to be multinomial as in discrete Bayesian networks. The second form only replaces each continuous variable whose descendants include discrete variables by a partner discrete variable a...

Joint distributions over many variables are frequently modeled by decomposing them into products of simpler, lower-dimensional conditional distributions, such as in sparsely connected Bayesian networks. However, automatically learning such models can be very computationally expensive when there are many datapoints and many continuous variables with complex nonlinear relationships, particularly when no good ways of decomposing the joint distribution are known a priori. In such situations, previous research has generally focused on the use of discretization techniques in which each continuous variable has a single discretization that is used throughout the entire network. In this paper, we present and compare a wide variety of tree-based algorithms for learning and evaluating conditional density estimates over continuous variables. These trees can be thought of as discretizations that vary according to the particular interactions being modeled; however, the density within a given leaf of the tree need not be assumed constant, and we show that such nonuniform leaf densities lead to more accurate density estimation. We have developed Bayesian network structure-learning algorithms that employ these tree-based conditional density representations, and we show that they can be used to practically learn complex joint probability models over dozens of continuous variables from thousands of datapoints. We focus on nding models that are simultaneously accurate, fast to learn, and fast to evaluate once they are learned.

no. 2002*H030300*000, and the University of Pittsburgh under contract no. 600322-1.