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：经验似然的贝叶斯推断面临一个挑战：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。该方法采用新颖的正则化技术消除凸包约束，通过引入连续指数族分布来满足库尔贝克-莱布勒散度准则。该正则化源于一种极限过程，在此过程中以结构化方式向指数倾斜经验似然的公式中添加伪数据。我们证明该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，并具有改进的有限样本表现。模拟与数据分析表明，所提方法为贝叶斯推断提供了合适的伪似然。该方法的实现可通过R包retel获取。本文的补充材料在线提供。