  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) and the sliced average variance estimation
(SAVE) type of methods, which is to use the statistical information of the
conditional distribution $\pi(\-x|y)$ to identify the dimension reduction (DR)
space. In particular we focus on the task of computing this conditional
distribution without slicing the data. 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 computed directly via Monte Carlo sampling. We then
can perform DR by considering certain moment functions (e.g. the first or the
second moment) of the samples of the posterior distribution. With numerical
examples, we demonstrate that the proposed method is especially effective for
small data problems.
