Scalable and accurate variational Bayes for high-dimensional binary regression models

Bayesian analysis of binary and polychotomous response data

Augusto Fasano, Daniele Durante and Giacomo Zanella

Biometrika, 2022, vol. 109, issue 4, 901-919

Abstract: SummaryModern methods for Bayesian regression beyond the Gaussian response setting are often computationally impractical or inaccurate in high dimensions. In fact, as discussed in recent literature, bypassing such a trade-off is still an open problem even in routine binary regression models, and there is limited theory on the quality of variational approximations in high-dimensional settings. To address this gap, we study the approximation accuracy of routinely used mean-field variational Bayes solutions in high-dimensional probit regression with Gaussian priors, obtaining novel and practically relevant results on the pathological behaviour of such strategies in uncertainty quantification, point estimation and prediction. Motivated by these results, we further develop a new partially factorized variational approximation for the posterior distribution of the probit coefficients that leverages a representation with global and local variables but, unlike for classical mean-field assumptions, it avoids a fully factorized approximation, and instead assumes a factorization only for the local variables. We prove that the resulting approximation belongs to a tractable class of unified skew-normal densities that crucially incorporates skewness and, unlike for state-of-the-art mean-field solutions, converges to the exact posterior density as . To solve the variational optimization problem, we derive a tractable coordinate ascent variational inference algorithm that easily scales toin the tens of thousands, and provably requires a number of iterations converging toas . Such findings are also illustrated in extensive empirical studies where our novel solution is shown to improve the approximation accuracy of mean-field variational Bayes for anyand , with the magnitude of these gains being remarkable in those high-dimensional n$]]> settings where state-of-the-art methods are computationally impractical.

Keywords: Bayesian computation; Data augmentation; High-dimensional probit regression; Truncated normal distribution; Unified skew-normal distribution; Variational Bayes (search for similar items in EconPapers)
Date: 2022
References: Add references at CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://hdl.handle.net/10.1093/biomet/asac026 (application/pdf)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:oup:biomet:v:109:y:2022:i:4:p:901-919.

Ordering information: This journal article can be ordered from
https://academic.oup.com/journals

Access Statistics for this article

Biometrika is currently edited by Paul Fearnhead

More articles in Biometrika from Biometrika Trust Oxford University Press, Great Clarendon Street, Oxford OX2 6DP, UK.
Bibliographic data for series maintained by Oxford University Press ().