# Multilevel Delayed Acceptance MCMC

arXiv: https://arxiv.org/abs/2202.03876 · HTML: https://mlsanity.com/paper/2202.03876/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2022.

## Built on by (most cited first, 10 of 10)

- [Multi-fidelity Gaussian process surrogate modeling for regression problems in physics](https://mlsanity.com/paper/2404.11965/) (2024)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Multi-Fidelity Delayed Acceptance: hierarchical MCMC sampling for Bayesian inverse problems combining multiple solvers through deep neural networks](https://mlsanity.com/paper/2512.16430/) (2025)
- [Fast-Mixing Markov Chains without Gradients](https://mlsanity.com/paper/2606.27564/) (2026)
- [Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models](https://mlsanity.com/paper/2212.03366/) (2022)
- [Accelerating Multilevel Markov Chain Monte Carlo Using Machine Learning Models](https://mlsanity.com/paper/2405.11179/) (2024)
- [Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators](https://mlsanity.com/paper/2410.20331/) (2024)
- [Efficient MCMC Sampling with Expensive-to-Compute and Irregular Likelihoods](https://mlsanity.com/paper/2505.10448/) (2025)
- [FLARE MCMC: Fidelity-based Layer-Adaptive REcursive proposals for MCMC](https://mlsanity.com/paper/2608.13774/) (2026)
- [BINDER: A Latent Variable Model for Probabilistic Medical Image Registration](https://mlsanity.com/paper/2609.19875/) (2026)

## Abstract (arXiv)

We develop a novel Markov chain Monte Carlo (MCMC) method that exploits a hierarchy of models of increasing complexity to efficiently generate samples from an unnormalized target distribution. Broadly, the method rewrites the Multilevel MCMC approach of Dodwell et al. (2015) in terms of the Delayed Acceptance (DA) MCMC of Christen & Fox (2005). In particular, DA is extended to use a hierarchy of models of arbitrary depth, and allow subchains of arbitrary length. We show that the algorithm satisfies detailed balance, hence is ergodic for the target distribution. Furthermore, multilevel variance reduction is derived that exploits the multiple levels and subchains, and an adaptive multilevel correction to coarse-level biases is developed. Three numerical examples of Bayesian inverse problems are presented that demonstrate the advantages of these novel methods. The software and examples are available in PyMC3.