On The Bayesian Multiple Index Models

Abstract

In modern statistical applications when the dimension is relatively large, it is a common practice to reduce the dimension using methods such as principal component analysis (PCA), sliced inverse regression and others before applying any statistical models. In this article, we synthetically combine these two steps by considering three Bayesian multi-index models: Bayesian multi-index additive model (BMIAM) for continuous response variable, Bayesian single-index model for binary response variable, and Bayesian multi-index model for categorical response variable. The indexes are parametrized by the hyper-spherical coordinates. The ridge functions are modeled using the Bayesian B-splines, which could be easily extended to other non-parametric methods. We have shown that the posterior consistency holds under certain conditions for the BMIAM. Further, we have developed the Markov chain Monte Carlo (MCMC) algorithm to sample the posterior of the proposed methods. It has been demonstrated through both simulation and real data analysis that the proposed methods provide a reliable estimation of indexes, dimension reduction space and good predictions for the responses.