Abstract Variational inference is a popular method for estimating model parameters and conditional distributions in hierarchical and mixed models, which arise frequently in many settings in the health, social, and biological sciences. Variational inference in a frequentist context works by approximating intractable conditional distributions with a tractable family and optimizing the resulting lower bound on the log-likelihood. The variational objective function is typically less computationally intensive to optimize than the true likelihood, enabling scientists to fit rich models even with extremely large datasets. Despite widespread use, little is known about the general theoretical properties of estimators arising from variational approximations to the log-likelihood, which hinders their use in inferential statistics. In this article, we connect such estimators to profile M-estimation, which enables us to provide regularity conditions for consistency and asymptotic normality of variational estimators. Our theory also motivates three methodological improvements to variational inference: estimation of the asymptotic model-robust covariance matrix, a one-step correction that improves estimator efficiency, and an empirical assessment of consistency. We evaluate the proposed results using simulation studies and data on marijuana use from the National Longitudinal Study of Youth. Supplementary materials for this article are available online.

中文翻译：

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超越预测：混合模型中的变分逼近推理框架

摘要 变分推理是一种流行的方法，用于估计分层模型和混合模型中的模型参数和条件分布，这在健康、社会和生物科学的许多环境中经常出现。频率论上下文中的变分推理通过用易处理的族逼近难处理的条件分布并优化对数似然的下界来工作。变分目标函数的优化计算量通常低于真实可能性，这使科学家能够拟合丰富的模型，即使是非常大的数据集。尽管被广泛使用，但对由对数似然变分近似产生的估计量的一般理论特性知之甚少，这阻碍了它们在推理统计中的使用。在本文中，我们将这些估计量连接到轮廓 M 估计，这使我们能够为变分估计量的一致性和渐近正态性提供正则性条件。我们的理论还激发了对变分推理的三个方法论改进：渐近模型稳健协方差矩阵的估计、提高估计器效率的一步校正以及一致性的经验评估。我们使用模拟研究和来自全国青年纵向研究的大麻使用数据来评估拟议的结果。本文的补充材料可在线获取。渐近模型稳健协方差矩阵的估计，提高估计器效率的一步校正，以及一致性的经验评估。我们使用模拟研究和来自全国青年纵向研究的大麻使用数据来评估拟议的结果。本文的补充材料可在线获取。渐近模型稳健协方差矩阵的估计，提高估计器效率的一步校正，以及一致性的经验评估。我们使用模拟研究和来自全国青年纵向研究的大麻使用数据来评估拟议的结果。本文的补充材料可在线获取。