Bayesian Restricted Likelihood, Generalized Bayes and Model Misspecification

Music City Center 

Model misspecification is problematic for Bayesians. Various methods have been proposed to modify the update from prior distribution to posterior distribution when one acknowledges that one's model is imperfect. Two current proposals are Bayesian restricted likelihood (BRL) methods and generalized Bayesian (GB) methods. The first focuses on aspects of the model that are believed to be modeled well and derives the posterior distribution by conditioning on an insufficient statistic (e.g., Huber's M-estimate) that captures those aspects of the model. The second focuses on a particular inference, making use of a loss function to define the target of inference. The usual Bayesian update is altered: the likelihood function is replaced with the exponentiated negative loss. We compare these two approaches by investigating both finite sample and asymptotic behavior when the data come from a location family. Suggestions for a choice between these two methods are given in different cases.

Partial Likelihood

Generalized Bayes Update

M-estimator

Posterior Asymptotics