An Efficient Bayesian Robust Principal Component Regression
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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).
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