Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.

翻译：结构化变分推断是现代统计学应用中的核心方法。与平均场变分推断不同，该方法假设近似后验具有相互依赖的结构。我们考虑星型结构变分推断的自然设定，其中一个根变量影响所有其他变量。我们首次证明了变分近似解的存在性、唯一性和自洽性结果。进而，我们推导了变分近似对后验分布的定量逼近误差界，将先前平均场设定的工作扩展至星型结构设定。我们还基于最优输运理论的思想，开发了一种具有可证明保证的梯度算法用于计算变分近似。我们探讨了所得结果对高斯测度与分层贝叶斯模型的启示，包括具有位置族先验的广义线性模型以及带有一维去偏的尖峰-厚板先验模型。作为分析的副产品，我们发展了星型可分离输运映射的新稳定性结果，这可能具有独立的学术价值。