  Sliced Inverse Regression (SIR) is an effective method for dimension
reduction in high-dimensional regression problems. The original method,
however, requires the inversion of the predictors covariance matrix. In case of
collinearity between these predictors or small sample sizes compared to the
dimension, the inversion is not possible and a regularization technique has to
be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it
is shown that SIR axes can be interpreted as solutions of an inverse regression
problem. In this paper, a Gaussian prior distribution is introduced on the
unknown parameters of the inverse regression problem in order to regularize
their estimation. We show that some existing SIR regularizations can enter our
framework, which permits a global understanding of these methods. Three new
priors are proposed leading to new regularizations of the SIR method. A
comparison on simulated data is provided.
