  Real-world problems, often couched as machine learning applications, involve
quantities of interest that have real-world meaning, independent of any
statistical model. To avoid potential model misspecification bias or
over-complicating the problem formulation, a direct, model-free approach is
desired. The traditional Bayesian framework relies on a model for the
data-generating process so, apparently, the desired direct, model-free,
posterior-probabilistic inference is out of reach. Fortunately, likelihood
functions are not the only means of linking data and quantities of interest.
Loss functions provide an alternative link, where the quantity of interest is
defined, or at least could be defined, as a minimizer of the corresponding
risk, or expected loss. In this case, one can obtain what is commonly referred
to as a Gibbs posterior distribution by using the empirical risk function
directly. This manuscript explores the Gibbs posterior construction, its
asymptotic concentration properties, and the frequentist calibration of its
credible regions. By being free from the constraints of model specification,
Gibbs posteriors create new opportunities for probabilistic inference in modern
statistical learning problems.
