Bayesian inverse regression for supervised dimension reduction with small datasets

We consider supervised dimension reduction problems, namely to identify a low dimensional projection of the predictors $\-x$ which can retain the statistical relationship between $\-x$ and the response variable $y$. We follow the idea of the sliced inverse regression (SIR) class of methods, which is to use the statistical information of the conditional distribution $\pi(\-x|y)$ to identify the dimension reduction (DR) space and in particular we focus on the task of computing this conditional distribution. We propose a Bayesian framework to compute the conditional distribution where the likelihood function is obtained using the Gaussian process regression model. The conditional distribution $\pi(\-x|y)$ can then be obtained directly by assigning weights to the original data points. We then can perform DR by considering certain moment functions (e.g. the first moment) of the samples of the posterior distribution. With numerical examples, we demonstrate that the proposed method is especially effective for small data problems.