# Importance is Important: Generalized Markov Chain Importance Sampling Methods

arXiv: https://arxiv.org/abs/2304.06251 · HTML: https://mlsanity.com/paper/2304.06251/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 57 references that are in the map.
- First posted on arXiv 13 Apr 2023.

## Builds on (most cited first, 9 of 9)

- [Oops I Took A Gradient: Scalable Sampling for Discrete Distributions](https://mlsanity.com/paper/2102.04509/) (2021)
- [The Barker proposal: combining robustness and efficiency in gradient-based MCMC](https://mlsanity.com/paper/1908.11812/) (2019)
- [On the Computational Complexity of High-Dimensional Bayesian Variable Selection](https://mlsanity.com/paper/1505.07925/) (2015)
- [The Hamming Ball Sampler](https://mlsanity.com/paper/1504.08133/) (2015)
- [Scalable Importance Tempering and Bayesian Variable Selection](https://mlsanity.com/paper/1805.00541/) (2018)
- [Markov Chain Importance Sampling -- a highly efficient estimator for MCMC](https://mlsanity.com/paper/1805.07179/) (2018)
- [Dimension-free Mixing for High-dimensional Bayesian Variable Selection](https://mlsanity.com/paper/2105.05719/) (2021)
- [Improving multiple-try Metropolis with local balancing](https://mlsanity.com/paper/2211.11613/) (2022)
- [Rapidly Mixing Multiple-try Metropolis Algorithms for Model Selection Problems](https://mlsanity.com/paper/2207.00689/) (2022)

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

- [From Minimax Optimal Importance Sampling to Uniformly Ergodic Importance-tempered MCMC](https://mlsanity.com/paper/2506.19186/) (2025)

## Abstract (arXiv)

We show that for any multiple-try Metropolis algorithm, one can always accept the proposal and evaluate the importance weight that is needed to correct for the bias without extra computational cost. This results in a general, convenient, and rejection-free Markov chain Monte Carlo (MCMC) sampling scheme. By further leveraging the importance sampling perspective on Metropolis--Hastings algorithms, we propose an alternative MCMC sampler on discrete spaces that is also outside the Metropolis--Hastings framework, along with a general theory on its complexity. Numerical examples suggest that the proposed algorithms are consistently more efficient than the original Metropolis--Hastings versions.