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We consider supervised dimension reduction problems, namely to identify a lowdimensional projection of the predictors $\-x$ which can retain the statisticalrelationship between $\-x$ and the response variable $y$. We follow the idea ofthe sliced inverse regression (SIR) and the sliced average variance estimation(SAVE) type of methods, which is to use the statistical information of theconditional distribution $\pi(\-x|y)$ to identify the dimension reduction (DR)space. In particular we focus on the task of computing this conditionaldistribution without slicing the data. We propose a Bayesian framework tocompute the conditional distribution where the likelihood function is obtainedusing the Gaussian process regression model. The conditional distribution$\pi(\-x|y)$ can then be computed directly via Monte Carlo sampling. We thencan perform DR by considering certain moment functions (e.g. the first or thesecond moment) of the samples of the posterior distribution. With numericalexamples, we demonstrate that the proposed method is especially effective forsmall data problems.

Bayesian inverse regression for dimension reduction with small datasets