Shrinkage priors for Bayesian penalized regression

In linear regression problems with many predictors, penalized regression techniques are often used to guard against overfitting and to select variables relevant for predicting an outcome variable. Recently, Bayesian penalization is becoming increasingly popular in which the prior distribution performs a function similar to that of the penalty term in classical penalization. Specifically, the so-called shrinkage priors in Bayesian penalization aim to shrink small effects to zero while maintaining true large effects. Compared to classical penalization techniques, Bayesian penalization techniques perform similarly or sometimes even better, and they offer additional advantages such as readily available uncertainty estimates, automatic estimation of the penalty parameter, and more flexibility in terms of penalties that can be considered. However, many different shrinkage priors exist and the available, often quite technical, literature primarily focuses on presenting one shrinkage prior and often provides comparisons with only one or two other shrinkage priors. This can make it difficult for researchers to navigate through the many prior options and choose a shrinkage prior for the problem at hand. Therefore, the aim of this paper is to provide a comprehensive overview of the literature on Bayesian penalization. We provide a theoretical and conceptual comparison of nine different shrinkage priors and parametrize the priors, if possible, in terms of scale mixture of normal distributions to facilitate comparisons. We illustrate different characteristics and behaviors of the shrinkage priors and compare their performance in terms of prediction and variable selection in a simulation study. Additionally, we provide two empirical examples to illustrate the application of Bayesian penalization. Finally, an R package bayesreg is available online (https://github.com/sara-vanerp/bayesreg) which allows researchers to perform Bayesian penalized regression with novel shrinkage priors in an easy manner.