Electronic Journal of Statistics

Consistency of variational Bayes inference for estimation and model selection in mixtures

Badr-Eddine Chérief-Abdellatif and Pierre Alquier

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Abstract

Mixture models are widely used in Bayesian statistics and machine learning, in particular in computational biology, natural language processing and many other fields. Variational inference, a technique for approximating intractable posteriors thanks to optimization algorithms, is extremely popular in practice when dealing with complex models such as mixtures. The contribution of this paper is two-fold. First, we study the concentration of variational approximations of posteriors, which is still an open problem for general mixtures, and we derive consistency and rates of convergence. We also tackle the problem of model selection for the number of components: we study the approach already used in practice, which consists in maximizing a numerical criterion (the Evidence Lower Bound). We prove that this strategy indeed leads to strong oracle inequalities. We illustrate our theoretical results by applications to Gaussian and multinomial mixtures.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 2995-3035.

Dates
Received: May 2018
First available in Project Euclid: 19 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1537344604

Digital Object Identifier
doi:10.1214/18-EJS1475

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 65C60: Computational problems in statistics 62F15: Bayesian inference 62F35: Robustness and adaptive procedures

Keywords
Mixture models frequentist evaluation of Bayesian methods variational approximations model selection

Rights
Creative Commons Attribution 4.0 International License.

Citation

Chérief-Abdellatif, Badr-Eddine; Alquier, Pierre. Consistency of variational Bayes inference for estimation and model selection in mixtures. Electron. J. Statist. 12 (2018), no. 2, 2995--3035. doi:10.1214/18-EJS1475. https://projecteuclid.org/euclid.ejs/1537344604


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Supplemental materials

  • Supplement to “Consistency of variational Bayes inference for estimation and model selection in mixtures”. The supplementary material zip contains the description of a short simulation study (supplement.pdf) and the notebook used for the simulation study (supplement.ipynb).