For many applications, a probability model can be more easily expressed as a cumulative distribution functions (CDF) as compared to the use of probability density or mass functions (PDF/PMFs). One advantage of CDF models is the simplicity of representing multivariate heavy-tailed distributions. Examples of fields that can benefit from the use of graphical models for CDFs include climatology and epidemiology, where datafollowheavy-taileddistributions and exhibit spatial correlations so that dependencies between model variables must be accounted for. However, in most cases the problem of learning from data consists of optimizing the log-likelihood function with respect to model parameters where we are required to optimize a log-PDF/PMF and not a log-CDF. Given a CDF defined on a graph, we present a message-passing algorithm called the gradient-derivative-product (GDP) algorithm that allows us to learn the model in terms of the log-likelihood function whereby messages correspond to local gradients of the likelihood with respect to model parameters. We demonstrate the GDP algorithm on real-world rainfall and H1N1 mortality data and weshow that the heavy-tailed multivariate distributions that arise in these problems can both be naturally parameterized and tractably estimated from data using our algorithm. 1

Directed acyclic graphs (DAGs) are a popular framework to express multivariate probability distributions. Acyclic directed mixed graphs (ADMGs) are generalizations of DAGs that can succinctly capture much richer sets of conditional independencies, and are especially useful in modeling the effects of latent variables implicitly. Unfortunately, there are currently no parameterizations of general ADMGs. In this paper, we apply recent work on cumulative distribution networks and copulas to propose one general construction for ADMG models. We consider a simple parameter estimation approach, and report some encouraging experimental results. MGs are. Reading off independence constraints from a ADMG can be done with a procedure essentially identical to d-separation (Pearl, 1988, Richardson and Spirtes, 2002). Given a graphical structure, the challenge is to provide a procedure to parameterize models that correspond to the independence constraints of the graph, as illustrated below. Example 1: Bi-directed edges correspond to some hidden common parent that has been marginalized. In the Gaussian case, this has an easy interpretation as constraints in the marginal covariance matrix of the remaining variables. Consider the two graphs below.