CONVERGENCE RATES OF VARIATIONAL POSTERIOR DISTRIBUTIONS

• 成果类型: Article
• 署名作者: Zhang, Fengshuo; Gao, Chao
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1883
• 发表日期: 2020
• 页码: 2180-2207
• 关键词: DENSITY-ESTIMATION; inference; contraction; likelihood; regression

摘要

We study convergence rates of variational posterior distributions for non-parametric and high-dimensional inference. We formulate general conditions on prior, likelihood and variational class that characterize the convergence rates. Under similar prior mass and testing conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by the variational approximation error. For a class of priors that admit the structure of a mixture of product measures, we propose a novel prior mass condition, under which the variational approximation error of the mean-field class is dominated by convergence rate of the true posterior. We demonstrate the applicability of our general results for various models, prior distributions and variational classes by deriving convergence rates of the corresponding variational posteriors.