# On Representations of Mean-Field Variational Inference

arXiv: https://arxiv.org/abs/2210.11385 · HTML: https://mlsanity.com/paper/2210.11385/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 27 references that are in the map.
- First posted on arXiv 20 Oct 2022.

## Builds on (most cited first, 4 of 4)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [Mean-field Variational Inference via Wasserstein Gradient Flow](https://mlsanity.com/paper/2207.08074/) (2022)

## Built on by (most cited first, 2 of 2)

- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)
- [Gaussian mixture layers for neural networks](https://mlsanity.com/paper/2508.04883/) (2025)

## Abstract (arXiv)

The mean field variational inference (MFVI) formulation restricts the general Bayesian inference problem to the subspace of product measures. We present a framework to analyze MFVI algorithms, which is inspired by a similar development for general variational Bayesian formulations. Our approach enables the MFVI problem to be represented in three different manners: a gradient flow on Wasserstein space, a system of Fokker-Planck-like equations and a diffusion process. Rigorous guarantees are established to show that a time-discretized implementation of the coordinate ascent variational inference algorithm in the product Wasserstein space of measures yields a gradient flow in the limit. A similar result is obtained for their associated densities, with the limit being given by a quasi-linear partial differential equation. A popular class of practical algorithms falls in this framework, which provides tools to establish convergence. We hope this framework could be used to guarantee convergence of algorithms in a variety of approaches, old and new, to solve variational inference problems.