# α-Variational Inference with Statistical Guarantees

arXiv: https://arxiv.org/abs/1710.03266 · HTML: https://mlsanity.com/paper/1710.03266/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2017.

## Built on by (most cited first, 4 of 4)

- [Concentration of a sparse Bayesian model with Horseshoe prior in estimating high-dimensional precision matrix](https://mlsanity.com/paper/2406.14269/) (2024)
- [Bayesian Model Selection via Mean-Field Variational Approximation](https://mlsanity.com/paper/2312.10607/) (2023)
- [Symmetry Guarantees Statistic Recovery in Variational Inference](https://mlsanity.com/paper/2604.18310/) (2026)
- [Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors](https://mlsanity.com/paper/2309.06349/) (2023)

## Abstract (arXiv)

We propose a family of variational approximations to Bayesian posterior distributions, called α-VB, with provable statistical guarantees. The standard variational approximation is a special case of α-VB with α=1. When α ∈(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 α-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, mixture of Gaussian models, latent Dirichlet allocation, and (mixture of) Gaussian variational approximation in regular parametric models.