  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 and propose a new method as an alternative. When working
with the new method, the data is summarized through a set of insufficient
statistics, targeting inferential quantities of interest, and the prior
distribution is updated with the summary statistics rather than the complete
data. By careful 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 captured by the conditioning statistics. For reducing
sensitivity to outliers, classical robust estimators (e.g., M-estimators) are
natural choices for conditioning statistics. A major contribution of this work
is the development of a data augmented Markov chain Monte Carlo (MCMC)
algorithm for the linear model and a large class of summary statistics. We
demonstrate the method on simulated and real data sets containing outliers and
subject to model misspecification. Success is manifested in better predictive
performance for data points of interest as compared to competing methods.
