# Geometric convergence of elliptical slice sampling

arXiv: https://arxiv.org/abs/2105.03308 · HTML: https://mlsanity.com/paper/2105.03308/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 26 references that are in the map.
- First posted on arXiv 7 May 2021.

## Builds on (most cited first, 3 of 3)

- [Perturbation theory for Markov chains via Wasserstein distance](https://mlsanity.com/paper/1503.04123/) (2015)
- [The Boomerang Sampler](https://mlsanity.com/paper/2006.13777/) (2020)
- [Efficient sampling for Gaussian linear regression with arbitrary priors](https://mlsanity.com/paper/1806.05738/) (2018)

## Built on by (most cited first, 5 of 5)

- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [ESS-Flow: Training-free guidance of flow-based models as inference in source space](https://mlsanity.com/paper/2510.05849/) (2025)
- [Reversibility of elliptical slice sampling revisited](https://mlsanity.com/paper/2301.02426/) (2023)
- [Gibbsian polar slice sampling](https://mlsanity.com/paper/2302.03945/) (2023)
- [Exploiting Concavity Information in Gaussian Process Contextual Bandit Optimization](https://mlsanity.com/paper/2503.10836/) (2025)

## Abstract (arXiv)

For Bayesian learning, given likelihood function and Gaussian prior, the elliptical slice sampler, introduced by Murray, Adams and MacKay 2010, provides a tool for the construction of a Markov chain for approximate sampling of the underlying posterior distribution. Besides of its wide applicability and simplicity its main feature is that no tuning is necessary. Under weak regularity assumptions on the posterior density we show that the corresponding Markov chain is geometrically ergodic and therefore yield qualitative convergence guarantees. We illustrate our result for Gaussian posteriors as they appear in Gaussian process regression, as well as in a setting of a multi-modal distribution. Remarkably, our numerical experiments indicate a dimension-independent performance of elliptical slice sampling even in situations where our ergodicity result does not apply.