Nonparametric Bayesian Modelling

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Nonparametric Bayesian Modelling Nonparametric Bayesian models are motivated by the desire to more realistically model data. They provide a means of escaping the strictures of parametric models, while, with their Bayesian nature, they allow the incorporation of some information about the process under investigation. These philosophical advantages translate directly into superior performance for the models, which have had a particularly strong showing wherever random effects are involved. When these models are used in conjunction with the hierarchical model, they constitute a powerful modelling tool. The current state of the art in nonparametric Bayesian modelling allows one to propose and fit models, and a limited amount of work has been done on selection of a model from some small set of candidate models. The greater field of data analysis in this context is almost untouched. The reason for this is that the current set of models do not allow for a sophisticated data analysis: the iterative process of proposing a model, assessing its fit, developing a modification of the model to improve its fit, and reducing the model if no substantial improvement in fit is found. The main focus of this work is to formulate and implement sophisticated nonparametric Bayesian data analysis. To accomplish the above goal, this research proposal identifies five areas where nonparametric Bayesian models currently either perform poorly or cannot be used in a satisfactory, general fashion. These areas are (i) examination of the relationship between covariates and response (ii) assessment of the fit of a model (iii) combination of information from related experiments (iv) modelling distributions with outliers and (v) working with small to moderate sample sizes. In the proposed research, the author will develop models that will show strong performance for each of the above problems. The first class of models, dependent nonparametric processes, move beyond current models which are aimed at providing a description of a single random distribution. They provide a means of modelling a collection of random distributions which exhibit strong local dependence and long-range independence. This feature of the model makes it ideal for (i) - (iii) above. The second class of models, contaminated models, directly targets the departure from a parametric model in a clear, easily interpretable fashion. This type of model is ideal for describing distributions that contain outliers, and its close proximity to a parametric form will yield small sample performance that is nearly equivalent to the parametric model, providing an approach to problems (iv) and (v). Additionally, the two types of models may be freely combined, resulting in a single, coherent approach to all five problems. The basic theoretical properties of the models will be investigated, the computational strategies needed to fit the models will be devised, and the models will be applied in a variety of settings. Throughout, the emphasis of the research will be on the development of sound data analytic strategies so that the theoretical and practical advantages of nonparametric Bayesian modelling can be realized.