# Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/2501.04259 · HTML: https://mlsanity.com/paper/2501.04259/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 8 Jan 2025.

## Builds on (most cited first, 5 of 5)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning](https://mlsanity.com/paper/2302.03271/) (2023)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Modeling, simulation and validation of supersonic parachute inflation dynamics during Mars landing](https://mlsanity.com/paper/1912.01658/) (2019)

## Built on by (most cited first, 3 of 3)

- [New affine invariant ensemble samplers and their dimensional scaling](https://mlsanity.com/paper/2505.02987/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)
- [Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems](https://mlsanity.com/paper/2605.29373/) (2026)

## Abstract (arXiv)

This paper is concerned with the approximation of probability distributions known up to normalization constants, with a focus on Bayesian inference for large-scale inverse problems in scientific computing. In this context, key challenges include costly repeated evaluations of forward models, multimodality, and inaccessible gradients for the forward model. To address them, we develop a variational inference framework that combines Fisher-Rao natural gradient with specialized quadrature rules to enable derivative free updates of Gaussian mixture variational families. The resulting method, termed Derivative Free Gaussian Mixture Variational Inference (DF-GMVI), guarantees covariance positivity and affine invariance, offering a stable and efficient framework for approximating complex posterior distributions. The effectiveness of DF-GMVI is demonstrated through numerical experiments on challenging scenarios, including distributions with multiple modes, infinitely many modes, and curved modes in spaces with up to 100 dimensions. The method's practicality is further demonstrated in a large-scale application, where it successfully recovers the initial conditions of the Navier-Stokes equations from solution data at positive times.