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Construction and Properties of Bayesian Nonparametric Regression Models


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6th August 2007 to 10th August 2007

Organisers: Professor Nils Hjort (Oslo), Dr Chris Holmes (Oxford), Professor Peter Müller (Texas) and Professor Stephen Walker (Canterbury)

Workshop Theme

Modern Bayesian nonparametric analysis was introduced by Ferguson with the development of the Dirichlet process in the 1970s. Since then there has been rapid progress in both theoretical and applied work, the latter usually relying on Markov chain Monte Carlo simulation methods. The outstanding challenges include the construction and properties, such as consistency issues, of Bayesian nonparametric regression models. This is the theme of the workshop.

Bayesian nonparametric inference relies on the construction of an infinite dimensional probability distribution on function spaces. Typically this is a space of density functions, but could also be hazard rate functions, distribution functions or some other function related to modelling observations. The probability models have traditionally been adapted from stochastic process; such as Lévy processes. These probabilities act as the prior distribution and combine with the data to provide the posterior distribution.

Another strand of Bayesian nonparametric inference involves the construction of regression functions. This is concerned with how observables relate to each other, where one type of observable (the predictor observable) is used to predict another type (the dependent observable). To date the functions used to do this have been quite different to those used for modeling density functions. For example, splines and wavelets are recent popular choices, and these have been used to model means and variances sitting within a class of parametric density functions.

Recent attempts have been made to strengthen the connections between these two area of Bayesian nonparametric research. The ultimate goal is obtain suitable classes of fully Bayesian nonparametric regression models. The current standard is semi-parametric; where if density functions are modeled nonparametrically, the regression function is modeled parametrically, and if the regression model is nonparametric then the density function which carries it is modeled parametrically.

The full Bayesian nonparametric regression model would involve the construction of a collection of density functions; one for each distinct predictor variable. The key question is how to connect up all the density functions to provide suitable interactions.

Leading international researchers in Bayesian nonparametric methods, both theory and applied, will be brought together to discuss and tackle this outstanding issue in Bayesian research.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons