Bayesian Restricted Likelihood Methods

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems (e.g., use of heavy-tailed likelihoods) and propose the restricted likelihood as an alternative. When working with restricted likelihood, we summarize the data through a set of (insufficient) statistics, targeting infer-ential quantities of interest, and update the prior distribution with the summary statistics rather than the complete data. By choice of conditioning statistics, we retain the main benefits of Bayesian methods while reducing the sensitivity of the analysis to features of the data not picked up by the conditioning statistics. A major contribution is the development of a data augmented MCMC algorithm for the linear model and a wide range of choices for summary statistics. 1 We demonstrate the method on an insurance agency data set containing many outliers and subject to model misspecification. Success is manifested in better predictive performance for data points of interest as compared to competing methods.