Bayesian Restricted Likelihood Methods: Conditioning on Insufficient Statistics in Bayesian Regression (with Discussion)

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and many well-documented advantages over competing methods. However, these run into difficulties for two major prevalent classes problems: handling data sets with outliers dealing model misspecification. We outline the drawbacks previous solutions both propose new method as an alternative. When working method, is summarized through set insufficient statistics, targeting inferential quantities interest, prior distribution updated summary statistics rather than complete data. By careful choice conditioning we retain main benefits while reducing sensitivity analysis features not captured by statistics. For outliers, classical robust estimators (e.g., M-estimators) are natural choices A contribution this work development augmented Markov chain Monte Carlo (MCMC) algorithm linear large class demonstrate on simulated real containing subject Success manifested in better predictive performance points interest compared