IDEAS home Printed from https://ideas.repec.org/a/taf/jnlasa/v114y2019i527p1147-1161.html
   My bibliography  Save this article

Frequentist Consistency of Variational Bayes

Author

Listed:
  • Yixin Wang
  • David M. Blei

Abstract

A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) methods. VB methods tend to be faster while achieving comparable predictive performance. However, there are few theoretical results around VB. In this article, we establish frequentist consistency and asymptotic normality of VB methods. Specifically, we connect VB methods to point estimates based on variational approximations, called frequentist variational approximations, and we use the connection to prove a variational Bernstein–von Mises theorem. The theorem leverages the theoretical characterizations of frequentist variational approximations to understand asymptotic properties of VB. In summary, we prove that (1) the VB posterior converges to the Kullback–Leibler (KL) minimizer of a normal distribution, centered at the truth and (2) the corresponding variational expectation of the parameter is consistent and asymptotically normal. As applications of the theorem, we derive asymptotic properties of VB posteriors in Bayesian mixture models, Bayesian generalized linear mixed models, and Bayesian stochastic block models. We conduct a simulation study to illustrate these theoretical results. Supplementary materials for this article are available online.

Suggested Citation

  • Yixin Wang & David M. Blei, 2019. "Frequentist Consistency of Variational Bayes," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1147-1161, July.
  • Handle: RePEc:taf:jnlasa:v:114:y:2019:i:527:p:1147-1161
    DOI: 10.1080/01621459.2018.1473776
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/01621459.2018.1473776
    Download Restriction: Access to full text is restricted to subscribers.
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yong Song & Tomasz Wo'zniak, 2020. "Markov Switching," Papers 2002.03598, arXiv.org.
    2. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    3. Ye Chen & Ilya O. Ryzhov, 2020. "Technical Note—Consistency Analysis of Sequential Learning Under Approximate Bayesian Inference," Operations Research, INFORMS, vol. 68(1), pages 295-307, January.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:jnlasa:v:114:y:2019:i:527:p:1147-1161. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Chris Longhurst). General contact details of provider: http://www.tandfonline.com/UASA20 .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.