Bayesian Variable Selection with the Quasi-Posterior

The Bayesian approach provides powerful methods for variable selection. The ability to incorporate sparsity through prior beliefs and account for parameter uncertainty allows Bayesian variable selection to consistently identify which variables are active and exhibit strong finite-sample performance. However, Bayesian methods require the correct specification of full likelihoods for the data, and there is increasing awareness of the problems that model misspecification causes for variable selection. Current approaches to mitigate misspecification either require complex models, detracting from the interpretability of the variable selection task, or move outside rigorous Bayesian uncertainty quantification and provide no recognised method for variable selection. This paper establishes the model quasi-posterior as a principled tool for variable selection.