CONVERGENCE RATES OF VARIATIONAL POSTERIOR DISTRIBUTIONS
CONVERGENCE RATES OF VARIATIONAL POSTERIOR DISTRIBUTIONS
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DOI：
10.1214/19-aos1883
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发表时间：
2020-08-01
影响因子：
4.5
通讯作者：
Gao, Chao
中科院分区：
文献类型：
--
作者：
Zhang, Fengshuo;Gao, Chao
我们研究了非参数和高维推断的变分后验分布的收敛速度。我们给出了刻画收敛速度的先验类、似然类和变分类的一般条件。在文献中考虑的类似的“先验质量和测试”条件下，速率被发现是两个项的总和。第一项代表真实后验分布的收敛速度，第二项由变分逼近误差贡献。对于一类允许混合乘积测度结构的先验，我们提出了一种新的先验质量条件，在该条件下，平均场类的变分逼近误差由真实后验的收敛速度控制。我们通过推导相应的变分后验的收敛速度，证明了我们的一般结果对各种模型、先验分布和变分类的适用性。
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.