  Bayesian posterior distributions are widely used for inference, but their
dependence on a statistical model creates some challenges. In particular, there
may be lots of nuisance parameters that require prior distributions and
posterior computations, plus a potentially serious risk of model
misspecification bias. Gibbs posterior distributions, on the other hand, offer
direct, principled, probabilistic inference on quantities of interest through a
loss function, not a model-based likelihood. Here we provide simple sufficient
conditions for establishing Gibbs posterior concentration rates when the loss
function is of a sub-exponential type. We apply these general results in a
range of practically relevant examples, including mean regression, quantile
regression, and sparse high-dimensional classification. We also apply these
techniques in an important problem in medical statistics, namely, estimation of
a personalized minimum clinically important difference.
