  Bayesian inference with empirical likelihood faces a challenge as the
posterior domain is a proper subset of the original parameter space due to the
convex hull constraint. We propose a regularized exponentially tilted empirical
likelihood to address this issue. Our method removes the convex hull constraint
using a novel regularization technique, incorporating a continuous exponential
family distribution to satisfy a Kullback-Leibler divergence criterion. The
regularization arises as a limiting procedure where pseudo-data are added to
the formulation of exponentially tilted empirical likelihood in a structured
fashion. We show that this regularized exponentially tilted empirical
likelihood retains certain desirable asymptotic properties of (exponentially
tilted) empirical likelihood and has improved finite sample performance.
Simulation and data analysis demonstrate that the proposed method provides a
suitable pseudo-likelihood for Bayesian inference. The implementation of our
method is available as the R package retel. Supplementary materials for this
article are available online.
