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Summary Model selection is crucial both to high-dimensional learning and to inference for contemporary big data applications in pinpointing the best set of covariates among a sequence of candidate interpretable models. Most existing work implicitly assumes that the models are correctly specified or have fixed dimensionality, yet both model misspecification and high dimensionality are prevalent in practice. In this paper, we exploit the framework of model selection principles under the misspecified generalized linear models presented in Lv & Liu (2014), and investigate the asymptotic expansion of the posterior model probability in the setting of high-dimensional misspecified models. With a natural choice of prior probabilities that encourages interpretability and incorporates the Kullback–Leibler divergence, we suggest using the high-dimensional generalized Bayesian information criterion with prior probability for large-scale model selection with misspecification. Our new information criterion characterizes the impacts of both model misspecification and high dimensionality on model selection. We further establish the consistency of covariance contrast matrix estimation and the model selection consistency of the new information criterion in ultrahigh dimensions under some mild regularity conditions. Our numerical studies demonstrate that the proposed method enjoys improved model selection consistency over its main competitors.
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Bayesian model selection via mean-field variational approximation
Abstract This article considers Bayesian model selection via mean-field (MF) variational approximation. Towards this goal, we study the non-asymptotic properties of MF inference that allows latent variables and model misspecification. Concretely, we show a Bernstein–von Mises (BvM) theorem for the variational distribution from MF under possible model misspecification, which implies the distributional convergence of MF variational approximation to a normal distribution centring at the maximal likelihood estimator. Motivated by the BvM theorem, we propose a model selection criterion using the evidence lower bound (ELBO), and demonstrate that the model selected by ELBO tends to asymptotically agree with the one selected by the commonly used Bayesian information criterion (BIC) as the sample size tends to infinity. Compared to BIC, ELBO tends to incur smaller approximation error to the log-marginal likelihood (a.k.a. model evidence) due to a better dimension dependence and full incorporation of the prior information. Moreover, we show the geometric convergence of the coordinate ascent variational inference algorithm, which provides a practical guidance on how many iterations one typically needs to run when approximating the ELBO. These findings demonstrate that variational inference is capable of providing a computationally efficient alternative to conventional approaches in tasks beyond obtaining point estimates.
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- Award ID(s):
- 2210717
- PAR ID:
- 10499684
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- ISSN:
- 1369-7412
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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