Safe-Bayesian Generalized Linear Regression

We study generalized Bayesian inference under misspecification, i.e. when the model is 'wrong but useful'. Generalized Bayes equips the likelihood with a learning rate eta. We show that for generalized linear models (GLMs), eta-generalized Bayes concentrates around the best approximation of the truth within the model for specific eta not equal 1, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes.