  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 $\eta \neq 1$, 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.
