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α-VARIATIONAL INFERENCE WITH STATISTICAL GUARANTEES
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DOI：10.1214/19-AOS1827.png)
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摘要
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We provide statistical guarantees for a family of variational approximations to Bayesian posterior distributions, called alpha-VB, which has close connections with variational approximations of tempered posteriors in the literature. The standard variational approximation is a special case of alpha-VB with alpha = 1. When alpha is an element of (0, 1], a novel class of variational inequalities are developed for linking the Bayes risk under the variational approximation to the objective function in the variational optimization problem, implying that maximizing the evidence lower bound in variational inference has the effect of minimizing the Bayes risk within the variational density family. Operating in a frequentist setup, the variational inequalities imply that point estimates constructed from the alpha-VB procedure converge at an optimal rate to the true parameter in a wide range of problems. We illustrate our general theory with a number of examples, including the mean-field variational approximation to (low)-high-dimensional Bayesian linear regression with spike and slab priors, Gaussian mixture models and latent Dirichlet allocation.
Keyword:
Bayes risk
evidence lower bound
latent variable models
Renyi divergence
variational inference
期刊
IF:
3.7
论文数:
2.8K
被引数:
2.9W
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