Variational Bayes for fast and accurate empirical likelihood inference

We develop a fast and accurate approach to approximate posterior distributions in the Bayesian empirical likelihood framework. Bayesian empirical likelihood allows for the use of Bayesian shrinkage without specification of a full likelihood but is notorious for leading to several computational difficulties. By coupling the stochastic variational Bayes procedure with an adjusted empirical likelihood framework, the proposed method overcomes the intractability of both the exact posterior and the arising evidence lower bound objective, and the mismatch between the exact posterior support and the variational posterior support. The optimization algorithm achieves fast algorithmic convergence by utilizing the variational expected gradient of the log adjusted empirical likelihood function. We prove the consistency of the proposed approximate posterior distribution and an empirical likelihood analogue of the variational Bernstein-von-Mises theorem. Through several numerical examples, we confirm the accuracy and quick algorithmic convergence of our proposed method.我们提出了一种快速精准的方法，用于在贝叶斯经验似然框架（Bayesian empirical likelihood framework）下近似后验分布。贝叶斯经验似然方法可在无需指定完整似然函数的前提下应用贝叶斯收缩，但该方法因存在诸多计算难题而广为人诟病。本文所提方法将随机变分贝叶斯（stochastic variational Bayes）过程与调整后的经验似然框架相结合，既克服了精确后验分布与由此导出的证据下界目标的难解性，又解决了精确后验支撑域与变分后验支撑域之间的不匹配问题。该优化算法通过利用对数调整经验似然函数的变分期望梯度，实现了快速的算法收敛。我们证明了所提近似后验分布的一致性，以及变分伯恩斯坦-冯·米塞斯（variational Bernstein-von-Mises）定理的经验似然类比形式。通过多个数值算例，我们验证了所提方法的准确性与快速收敛特性。