1.1 Dimension reduction based on finite mixture modeling of inverse regression

Consider the usual regression problem in which we want to study the conditional distri- bution of a response Y given a set of predictors X. Sufficient dimension reduction (SDR) methods aim at replacing the high-dimensional vector of predictors by a lower-dimensional function R(X) with no loss of information about the dependence of the response variable on the predictors. Almost all SDR methods restrict attention to the class of linear reduc- tions, which can be represented in terms of the projection of X onto a dimension-reduction subspace (DRS). Several methods have been proposed to estimate the basis of the DRS, such as sliced inverse regression (SIR; Li, 1991), principal Hessian directions (PHD; Li, 1992), sliced average variance estimation (SAVE; Cook and Weisberg, 1991), directional regression (DR; Li et al., 2005) and inverse regression estimation (IRE; Cook and Ni, 2005). A novel SDR method, called MSIR, based on finite mixtures of Gaussians has been recently proposed (Scrucca, 2011) as an extension to SIR. The talk will present the MSIR methodology and some recent advances. In particular, a BIC criterion for the selection the dimensionality of DRS will be introduced, and its extension for the purpose of variable selection. Finally, the application of MSIR in classification problems, both supervised and semi-supervised, will be discussed.