Gibbs posterior inference on multivariate quantiles

Abstract

Bayesian and other likelihood-based methods require specification of a statistical model and may not be fully satisfactory for inference on quantities, such as quantiles, that are not naturally defined as model parameters. In this paper, we construct a direct and model-free Gibbs posterior distribution for multivariate quantiles. Being model-free means that inferences drawn from the Gibbs posterior are not subject to model misspecification bias, and being direct means that no priors for or marginalization over nuisance parameters are required. We show here that the Gibbs posterior enjoys a root-$n$ convergence rate and a Bernstein--von Mises property, i.e., for large n, the Gibbs posterior distribution can be approximated by a Gaussian. Moreover, we present numerical results showing the validity and efficiency of credible sets derived from a suitably scaled Gibbs posterior.

24 pages, 3 figures, 2 tables

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Cited by in corpus (4)

• A comparison of learning rate selection methods in generalized Bayesian inference
• Gibbs posterior concentration rates under sub-exponential type losses
• Bernstein - von Mises theorem and misspecified models: a review
• Statistical Inference for Bayesian Risk Minimization via Exponentially Tilted Empirical Likelihood