Statistical models based on acyclic directed graphs (ADGs) (also called directed acyclic graphs (DAGs), Bayesian networks, or influence diagrams) are particularly well behaved, easily interpretable, and computationally convenient. In the late 1980s, ADG models were generalized to adicyclic graphs or chain graphs, which include both directed and undirected edges, hence can simultaneously represent dependences some of which are directional and some associative. The investigators will study the Markov and statistical properties of a new class of chain graph models that retains many of the desirable properties of ADG models. Problems to be investigated include the completeness and faithfulness of these new models, determination of their local Markov property, and characterization of their Markov equivalence classes by means of an appropriate essential graph. The investigators will also study a very general class of Wishart distributions on homogeneous cones, in which transitive acyclic directed graphs (TADGs) play a central role. E. B. Vinberg's classical characterization of homogeneous cones has been found to reveal a fundamental relationship between normal models satisfying TADG Markov conditions and this general class of Wishart distributions. This class includes all Wishart distributions previously known in multivariate statistical analysis, including the hyper-Wishart distributions and Wishart distributions associated with normal lattice conditional independence (LCI) models, as well as a great many new ones. Additional topics to be investigated include the limitations of the Neyman-Pearson, likelihood ratio, and maximum likelihood criteria for multiparameter hypothesis-testing and estimation problems, and the efficacy of the likelihood ratio test for testing order-restricted and multivariate one-sided alternatives.
One of the most central ideas of statistical science is the assessment of dependences among a set of stochastic variables. The familiar concepts of correlation, regression, and prediction are manifestations of this idea, and many aspects of causal relationships ultimately rest on representations of multivariate dependence. Graphical Markov models (GMM) use graphs i.e. networks, either undirected, directed, or mixed, to represent multivariate dependencies in a visual and computationally efficient manner. A GMM is usually constructed by specifying local dependences for each variable, i.e. node of the graph, in terms of its immediate neighbors, parents, or both, yet can represent a highly varied and complex system of multivariate dependences by means of the global structure of the graph. The local specification permits efficiencies in modeling, inference, and probabilistic calculations. Among their many applications, GMMs have become prevalent in statistical science for the analysis of categorical data in contingency tables, for the modeling of spatially-dependent processes such as the spread of epidemics in human and animal populations, and for the development of early warning systems for severe weather conditions; in computer science (as Bayesian networks) for information processing and retrieval, for robotics, computer vision, and pattern recognition, for the debugging of complex programs (such as Windows 98), and for the representation of expert systems for medical diagnosis; and in decision science (as influence diagrams) as models for information flow and control and for combining the opinions of many decision-makers. A crucial feature of these models is that they are designed for fast computational implementation, thereby facilitating the development of software that can "reason" about real world problems.
