Title: An Efficient Bayesian Robust Principal Component Regression,
Abstract: Principal component regression is a linear regression model with principal
components as regressors. This type of modelling is particularly useful for
prediction in settings with high-dimensional covariates. Surprisingly, the
existing literature treating of Bayesian approaches is relatively sparse. In
this paper, we aim at filling some gaps through the following practical
contribution: we introduce a Bayesian approach with detailed guidelines for a
straightforward implementation. The approach features two characteristics that
we believe are important. First, it effectively involves the relevant principal
components in the prediction process. This is achieved in two steps. The first
one is model selection; the second one is to average out the predictions
obtained from the selected models according to model averaging mechanisms,
allowing to account for model uncertainty. The model posterior probabilities
are required for model selection and model averaging. For this purpose, we
include a procedure leading to an efficient reversible jump algorithm. The
second characteristic of our approach is whole robustness, meaning that the
impact of outliers on inference gradually vanishes as they approach plus or
minus infinity. The conclusions obtained are consequently consistent with the
majority of observations (the bulk of the data).