Collaborative Research: Graphical and Algebraic Models for Multivariate Categorical Data

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The proposed research project develops models for multivariate categoricaldata by mimicking Gaussian models with a desired model structure that canbe captured in terms of the non-parametric concept of conditionalindependence. This method has a long history: graphical log-linear modelscan be induced in this way by Gaussian models defined by zero constraintson the inverse covariance matrix. The project seeks to greatly extend thescope of the approach. It is proposed to define and study marginalindependence models for contingency tables, discrete-valued time serieswith moving average-like dependence structure, seemingly unrelatedregressions with discrete response variables, and discrete graphicalmodels based on the recently introduced AMP chain graphs and ancestralgraphs. The main objectives of the study are development ofparameterizations, construction and implementation of efficient algorithmsfor maximum likelihood estimation, and investigation of procedures formodel selection. A particular focus of the project will be on employingmodern tools from computational algebra in the analysis of the structure ofparameter spaces and properties of likelihood functions.Multivariate statistical models seek to describe the complex relationshipsbetween a large set of variables. A particular class of such models,called graphical models, has found wide-spread application in fields likeartificial intelligence, bio-informatics, biology, epidemiology, andspeech recognition. The models proposed in the project extend the realmof graphical models and it is anticipated that they will be applied inmany of these fields. Moreover, the proposed methodology will provide newtools for the analysis of data of public interest such as census data.The researchers also plan to make software tools freely available as partof a larger open source statistical software package called R.