Scaling the Gibbs posterior credible regions

Abstract

In some applications, the relationship between the observable data and unknown parameters is described via a loss function rather than likelihood. In such cases, the standard Bayesian methodology cannot be used, but a Gibbs posterior distribution can be constructed by appropriately using the loss in place of a likelihood. Although misspecified by definition, Gibbs posteriors have a number of desirable properties; for example, we show that the Gibbs posterior converges at the same rate as the corresponding M-estimator. Inference based on the Gibbs posterior is not straight- forward, however, because the finite-sample performance is highly sensitive to the scale of the loss function. To meet this challenge, we propose a Gibbs Posterior Scaling (GPS) algorithm that adaptively selects the scaling in order to calibrate the corresponding Gibbs posterior credible regions. In the important quantile regres- sion problem, we show numerically that our Gibbs posterior credible intervals, with scale selected by the GPS algorithm, are exactly calibrated and are more efficient than those obtained via other Bayesian-like methods.