  Generalized likelihoods are commonly used to obtain consistent estimators
with attractive computational and robustness properties. Formally, any
generalized likelihood can be used to define a generalized posterior
distribution, but an arbitrarily defined "posterior" cannot be expected to
appropriately quantify uncertainty in any meaningful sense. In this article, we
provide sufficient conditions under which generalized posteriors exhibit
concentration, asymptotic normality (Bernstein-von Mises), an asymptotically
correct Laplace approximation, and asymptotically correct frequentist coverage.
We apply our results in detail to generalized posteriors for a wide array of
generalized likelihoods, including pseudolikelihoods in general, the Gaussian
Markov random field pseudolikelihood, the fully observed Boltzmann machine
pseudolikelihood, the Ising model pseudolikelihood, the Cox proportional
hazards partial likelihood, and a median-based likelihood for robust inference
of location. Further, we show how our results can be used to easily establish
the asymptotics of standard posteriors for exponential families and generalized
linear models. We make no assumption of model correctness so that our results
apply with or without misspecification.
