提出星形结构变分推断的理论框架与高效计算方法。
Theory and computation for structured variational inference
- 基于最优传输思想设计可证明收敛的梯度算法
- 首次建立星形结构变分近似存在性、唯一性与自洽性
- 适用于高斯测度和层次贝叶斯模型，适合理论研究者
结构化变分推断是现代统计应用的核心方法。与均值场变分推断不同，其近似后验假设变量间存在依赖结构。本文研究星形结构变分推断，即一个根变量影响其余所有变量的情形。首次证明了变分近似的存在性、唯一性和自洽性，并推导出该近似对后验分布的定量误差界，将先前均值场设定下的结果拓展至星形结构设定。同时，基于最优传输理论，提出一种具有理论保证的梯度算法来计算变分近似。探讨了这些结果在高斯测度和层次贝叶斯模型中的应用，包括位置族先验的广义线性模型以及一维去偏的spike-and-slab先验。作为分析副产品，还建立了星可分离传输映射的新稳定性结果，可能具有独立兴趣。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。