# Involutive MCMC: a Unifying Framework

arXiv: https://arxiv.org/abs/2006.16653 · HTML: https://mlsanity.com/paper/2006.16653/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2020.

## Built on by (most cited first, 19 of 19)

- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [MixFlows: principled variational inference via mixed flows](https://mlsanity.com/paper/2205.07475/) (2022)
- [Hamiltonian Dynamics with Non-Newtonian Momentum for Rapid Sampling](https://mlsanity.com/paper/2111.02434/) (2021)
- [Recursive Monte Carlo and Variational Inference with Auxiliary Variables](https://mlsanity.com/paper/2203.02836/) (2022)
- [Deep Learning Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.03418/) (2021)
- [Sequential Monte Carlo Learning for Time Series Structure Discovery](https://mlsanity.com/paper/2307.09607/) (2023)
- [Automatically Marginalized MCMC in Probabilistic Programming](https://mlsanity.com/paper/2302.00564/) (2023)
- [AutoStep: Locally adaptive involutive MCMC](https://mlsanity.com/paper/2410.18929/) (2024)
- [LeapfrogLayers: A Trainable Framework for Effective Topological Sampling](https://mlsanity.com/paper/2112.01582/) (2021)
- [Mixed Variational Flows for Discrete Variables](https://mlsanity.com/paper/2308.15613/) (2023)
- [Nonparametric Involutive Markov Chain Monte Carlo](https://mlsanity.com/paper/2211.01100/) (2022)
- [Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps](https://mlsanity.com/paper/2406.02490/) (2024)
- [Asymptotically exact variational flows via involutive MCMC kernels](https://mlsanity.com/paper/2506.02162/) (2025)
- [Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo](https://mlsanity.com/paper/2301.01409/) (2023)
- [Markov chain Monte Carlo without evaluating the target: an auxiliary variable approach](https://mlsanity.com/paper/2406.05242/) (2024)
- [Incremental Structure Discovery of Classification via Sequential Monte Carlo](https://mlsanity.com/paper/2408.07875/) (2024)
- [Bayesian Inverse Physics for Neuro-Symbolic Robot Learning](https://mlsanity.com/paper/2506.08756/) (2025)
- [Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling](https://mlsanity.com/paper/2607.15682/) (2026)
- [LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs](https://mlsanity.com/paper/2608.08588/) (2026)

## Abstract (arXiv)

Markov Chain Monte Carlo (MCMC) is a computational approach to fundamental problems such as inference, integration, optimization, and simulation. The field has developed a broad spectrum of algorithms, varying in the way they are motivated, the way they are applied and how efficiently they sample. Despite all the differences, many of them share the same core principle, which we unify as the Involutive MCMC (iMCMC) framework. Building upon this, we describe a wide range of MCMC algorithms in terms of iMCMC, and formulate a number of "tricks" which one can use as design principles for developing new MCMC algorithms. Thus, iMCMC provides a unified view of many known MCMC algorithms, which facilitates the derivation of powerful extensions. We demonstrate the latter with two examples where we transform known reversible MCMC algorithms into more efficient irreversible ones.