# Gaussian Invariant Markov Chain Monte Carlo

arXiv: https://arxiv.org/abs/2506.21511 · HTML: https://mlsanity.com/paper/2506.21511/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 26 Jun 2025.

## Builds on (most cited first, 7 of 7)

- [Auxiliary gradient-based sampling algorithms](https://mlsanity.com/paper/1610.09641/) (2016)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [Gradient-based Adaptive Markov Chain Monte Carlo](https://mlsanity.com/paper/1911.01373/) (2019)
- [Optimal Preconditioning and Fisher Adaptive Langevin Sampling](https://mlsanity.com/paper/2305.14442/) (2023)
- [Convergence of position-dependent MALA with application to conditional simulation in GLMMs](https://mlsanity.com/paper/2108.12662/) (2021)
- [Variance Reduction for Metropolis-Hastings Samplers](https://mlsanity.com/paper/2203.02268/) (2022)
- [Variance Reduction for Independent Metropolis](https://mlsanity.com/paper/2406.17699/) (2024)

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

- [Transformed Samplers with Variance Reduction](https://mlsanity.com/paper/2610.10870/) (2026)

## Abstract (arXiv)

We develop sampling methods, which consist of Gaussian invariant versions of random walk Metropolis (RWM), Metropolis adjusted Langevin algorithm (MALA) and second order Hessian or Manifold MALA. Unlike standard RWM and MALA, we show that Gaussian invariant sampling can lead to ergodic estimators with improved statistical efficiency. This is due to a remarkable property of Gaussian invariance that allows us to obtain exact analytical solutions to the Poisson equation for Gaussian targets. These solutions can be used to construct efficient and easy to use control variates for variance reduction of estimators under any intractable target. We demonstrate the new samplers and estimators in several examples, including high dimensional targets in latent Gaussian models where we compare against several advanced methods and obtain state-of-the-art results. We also provide theoretical results regarding geometric ergodicity, and an optimal scaling analysis that shows the dependence of the optimal acceptance rate on the Gaussianity of the target.