Near optimal prediction from relevant components with Bayes PLS

The theoretical PLS solution has been shown to stop automatically after the relevant number of components has been extracted (Helland, 1990). The sample PLS algorithm can be regarded as a plug-in procedure where population parameters are replaced by sample estimates. Theoretical arguments have been given that the sample PLS solution cannot be optimal for prediction since this estimator falls outside the parameter space of the theoretical PLS algorithm (Helland, 2001). We aim at developing a near optimal predictor from the population PLS model. It may be shown that the best equivariant estimator under certain symmetry assumptions is the Bayes estimator with certain invariant priors. The Bayes estimator is found using Markov Chain Monte Carlo methods and the method is illustrated on a small data set.