# Markov Chain Importance Sampling -- a highly efficient estimator for MCMC

arXiv: https://arxiv.org/abs/1805.07179 · HTML: https://mlsanity.com/paper/1805.07179/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 6 of 6)

- [MCMC-driven importance samplers](https://mlsanity.com/paper/2105.02579/) (2021)
- [Compressed Monte Carlo with application in particle filtering](https://mlsanity.com/paper/2107.08459/) (2021)
- [Proxy-informed Bayesian transfer learning with unknown sources](https://mlsanity.com/paper/2411.03263/) (2024)
- [Importance is Important: Generalized Markov Chain Importance Sampling Methods](https://mlsanity.com/paper/2304.06251/) (2023)
- [Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo](https://mlsanity.com/paper/2410.18993/) (2024)
- [From Minimax Optimal Importance Sampling to Uniformly Ergodic Importance-tempered MCMC](https://mlsanity.com/paper/2506.19186/) (2025)

## Abstract (arXiv)

Markov chain (MC) algorithms are ubiquitous in machine learning and statistics and many other disciplines. Typically, these algorithms can be formulated as acceptance rejection methods. In this work we present a novel estimator applicable to these methods, dubbed Markov chain importance sampling (MCIS), which efficiently makes use of rejected proposals. For the unadjusted Langevin algorithm, it provides a novel way of correcting the discretization error. Our estimator satisfies a central limit theorem and improves on error per CPU cycle, often to a large extent. As a by-product it enables estimating the normalizing constant, an important quantity in Bayesian machine learning and statistics.