Probabilistic graphical models use local factors to represent dependence among sets of variables. For many problem domains, for instance climatology and epidemiology, in addition to local dependencies, we may also wish to model heavy-tailed statistics, where extreme deviations should not be treated as outliers. Specifying such distributions using graphical models for probability density functions (PDFs) generally lead to intractable inference and learning. Cumulative distribution networks (CDNs) provide a means to tractably specify multivariate heavy-tailed models as a product of cumulative distribution functions (CDFs). Currently, algorithms for inference and learning, which correspond to computing mixed derivatives, are exact only for tree-structured graphs.

Low-dimensional probability models for local distribution functions in a Bayesian network include decision trees, decision graphs, and causal independence models. We describe a new probability model for discrete Bayesian networks, which we call an embedded Bayesian network classifier or EBNC. The model for a node $Y$ given parents $\bf X$ is obtained from a (usually different) Bayesian network for $Y$ and $\bf X$ in which $\bf X$ need not be the parents of $Y$. We show that an EBNC is a special case of a softmax polynomial regression model. Also, we show how to identify a non-redundant set of parameters for an EBNC, and describe an asymptotic approximation for learning the structure of Bayesian networks that contain EBNCs. Unlike the decision tree, decision graph, and causal independence models, we are unaware of a semantic justification for the use of these models. Experiments are needed to determine whether the models presented in this paper are useful in practice.

Probabilistic graphical models use local factors to represent dependence among sets of variables. For many problem domains, for instance climatology and epidemiology, in addition to local dependencies, we may also wish to model heavy-tailed statistics, where extreme deviations should not be treated as outliers. Specifying such distributions using graphical models for probability density functions (PDFs) generally lead to intractable inference and learning. Cumulative distribution networks (CDNs) provide a means to tractably specify multivariate heavy-tailed models as a product of cumulative distribution functions (CDFs). Currently, algorithms for inference and learning, which correspond to computing mixed derivatives, are exact only for tree-structured graphs. For graphs of arbitrary topology, an efficient algorithm is needed that takes advantage of the sparse structure of the model, unlike symbolic differentiation programs such as Mathematica and D* that do not. We present an algorithm for recursively decomposing the computation of derivatives for CDNs of arbitrary topology, where the decomposition is naturally described using junction trees. We compare the performance of the resulting algorithm to Mathematica and D*, and we apply our method to learning models for rainfall and H1N1 data, where we show that CDNs with cycles are able to provide a significantly better fits to the data as compared to tree-structured and unstructured CDNs and other heavy-tailed multivariate distributions such as the multivariate copula and logistic models.