Bayesian nonparametric single-index regression
Seminar Room 1, Newton Institute
The single-index model provides a flexible approach to nonlinear regression, and unlike many nonparametric regression models, this model is interpretable, easily handles high-dimensional covariates, and readily incorporates interactions among individual covariates. The model is defined by y = g(w'x) + e, where g is an unspecified univariate (ridge) function, x is a p-dimensional covariate, and the regression errors e,...,e[n] are assumed to be iid Normal with common variance. Previous research on the single-index model has mainly been limited to issues of point-estimation, where specifically, the focus is to estimate the function g by maximizing a penalized likelihood, with the function defined by splines having a fixed number of knots located at fixed locations of the predictor space. In this talk I will discuss a novel, Bayesian nonparametric approach to the single-index model, where the function g is modeled by linear splines with the number and locations of knots treated as unknown parameters, where random-effect parameters are added to the model to describe the effects of different clusters of observations, and where the variance of the regression error is allowed to change nonparametrically with the value of the covariate. In particular, the random-effects are modeled by a Dirichlet Process centered on a Normal distribution, and the error variances are modeled by a Dirichlet Process centered on a inverse-gamma distribution. Moreover, this new single-index model can readily handle observed dependent variables that are either continuous, binary, ordered-categories, or counts. I will illustrate Bayesian nonparametric single-index models through the analysis of real data of students from secondary schools.
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