Principal component regression is a linear regression model with principalcomponents as regressors. This type of modelling is particularly useful forprediction in settings with high-dimensional covariates. Surprisingly, theexisting literature treating of Bayesian approaches is relatively sparse. Inthis paper, we aim at filling some gaps through the following practicalcontribution: we introduce a Bayesian approach with detailed guidelines for astraightforward implementation. The approach features two characteristics thatwe believe are important. First, it effectively involves the relevant principalcomponents in the prediction process. This is achieved in two steps. The firstone is model selection; the second one is to average out the predictionsobtained from the selected models according to model averaging mechanisms,allowing to account for model uncertainty. The model posterior probabilitiesare required for model selection and model averaging. For this purpose, weinclude a procedure leading to an efficient reversible jump algorithm. Thesecond characteristic of our approach is whole robustness, meaning that theimpact of outliers on inference gradually vanishes as they approach plus orminus infinity. The conclusions obtained are consequently consistent with themajority of observations (the bulk of the data).
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