# Markov Chain Monte Carlo with Diffusion Paths

arXiv: https://arxiv.org/abs/2607.11631 · HTML: https://mlsanity.com/paper/2607.11631/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 79 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Simulated Tempering Langevin Monte Carlo II: An Improved Proof using Soft Markov Chain Decomposition](https://mlsanity.com/paper/1812.00793/) (2018)
- [Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching](https://mlsanity.com/paper/2603.00530/) (2026)
- [Diffusion-based Annealed Boltzmann Generators : benefits, pitfalls and hopes](https://mlsanity.com/paper/2601.21026/) (2026)

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

- [Exact Global MCMC with Denoising Diffusion](https://mlsanity.com/paper/2609.00279/) (2026)
- [Leveraging generative models to assist Monte Carlo sampling](https://mlsanity.com/paper/2608.07648/) (2026)
- [Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling](https://mlsanity.com/paper/2607.15682/) (2026)
- [Discrete Diffusion Models via Evolving Variational Autoregressive Networks](https://mlsanity.com/paper/2609.27306/) (2026)
- [Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler](https://mlsanity.com/paper/2609.28338/) (2026)

## Abstract (arXiv)

Sampling from multimodal distributions is a longstanding challenge for classical local Markov chain Monte Carlo (MCMC) methods. A popular remedy is to introduce a sequence of intermediate distributions that interpolate between the target and a simpler reference. The classical choice, tempering, raises the density to a power, but distorts the relative weights of asymmetric modes and can lead to poor mixing. We instead propose interpolating along the diffusion path, the marginals of a noising diffusion process that carries the target toward a Gaussian. This path preserves the relative weights of the modes and enjoys favorable mixing properties, which we make precise through a spectral-gap analysis of the corresponding ideal transition kernel. Sampling along the path requires its intermediate scores, which can be estimated from the unnormalized target through variational approaches, yielding only an approximate sampler. To remove the resulting bias, we introduce the Metropolis-adjusted diffusion path (MAD-Path) sampler, which corrects the diffusion-path proposal in an augmented path space and leaves the target invariant regardless of the accuracy of the learned score or the discretization error. We further quantify how these two errors affect the acceptance probability, providing guidance for practical tuning. Experiments on a range of Bayesian posteriors show that MAD-Path improves global exploration and mode-weight estimation relative to tempering-based MCMC methods and unadjusted diffusion samplers.