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.

翻译：贝叶斯推断结合经验似然时面临一个挑战：由于凸包约束，后验域是原始参数空间的真子集。我们提出一种正则化指数倾斜经验似然方法来解决此问题。该方法通过一种新颖的正则化技术移除凸包约束，引入连续指数族分布以满足Kullback-Leibler散度准则。该正则化源于一个极限过程，其中伪数据以结构化方式加入指数倾斜经验似然的公式中。我们证明这种正则化指数倾斜经验似然保留了（指数倾斜）经验似然的某些理想渐近性质，并改善了有限样本表现。模拟与数据分析表明，所提出的方法为贝叶斯推断提供了合适的伪似然函数。该方法的实现可获取为R包 retel。本文补充材料在线提供。