# Symmetry Guarantees Statistic Recovery in Variational Inference

arXiv: https://arxiv.org/abs/2604.18310 · HTML: https://mlsanity.com/paper/2604.18310/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 33 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Builds on (most cited first, 17 of 17)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [Markovian Score Climbing: Variational Inference with KL(p||q)](https://mlsanity.com/paper/2003.10374/) (2020)
- [Convergence Rates of Variational Posterior Distributions](https://mlsanity.com/paper/1712.02519/) (2017)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Black-box α-divergence Minimization](https://mlsanity.com/paper/1511.03243/) (2015)
- [On Statistical Optimality of Variational Bayes](https://mlsanity.com/paper/1712.08983/) (2017)
- [Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics](https://mlsanity.com/paper/2210.06226/) (2022)
- [Alpha-Beta Divergence For Variational Inference](https://mlsanity.com/paper/1805.01045/) (2018)
- [Statistical Inference in Mean-Field Variational Bayes](https://mlsanity.com/paper/1911.01525/) (2019)
- [Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix](https://mlsanity.com/paper/2410.11067/) (2024)
- [α-Variational Inference with Statistical Guarantees](https://mlsanity.com/paper/1710.03266/) (2017)
- [Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry](https://mlsanity.com/paper/2511.01064/) (2025)

## Built on by (most cited first, 1 of 1)

- [Gaussian Mean Field Variational Inference can Overestimate Predictive Variance](https://mlsanity.com/paper/2606.25745/) (2026)

## Abstract (arXiv)

Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.