A New Perspective on Reverse Diffusion for Monte Carlo Sampling

2026-06-02

AI Summary

This work addresses the challenge of efficient sampling from unnormalized densities, circumventing the time discretization errors and score estimation biases inherent in conventional approaches. By embedding the target distribution as the initial marginal of a finite-time reverse diffusion process and leveraging an Ornstein–Uhlenbeck process to construct the Radon–Nikodym derivative, the authors establish a measure transformation framework that requires neither discretization nor score estimation. Building on this foundation, they propose two algorithms: a fully parallel, approximately i.i.d. sampler and a path-space Markov chain Monte Carlo (MCMC) update mechanism, where acceptance probabilities are realized via a Barker-type Bernoulli factory. The unified framework demonstrates substantial improvements over random-walk Metropolis on multimodal and strongly dependent distributions, offering both scalability and computational efficiency.

Abstract

This paper introduces a novel perspective on the use of reverse diffusion processes for sampling from unnormalized densities. The central idea is to embed the target density as the marginal at the initial time of a suitably constructed diffusion process evolving over a finite horizon. In contrast to existing approaches, the proposed methodology involves neither time discretization error nor score function estimation, so that Monte Carlo variability is the only source of approximation. A key theoretical result characterizes the Radon-Nikodym derivative of the reverse diffusion transition distribution with respect to that of an Ornstein-Uhlenbeck (OU) process. This representation provides a tractable change-of-measure formulation and serves as the foundation for two distinct classes of Monte Carlo algorithms. The first class approximates the reverse transition distribution via a sequence of pseudo-marginal Metropolis-Hastings MCMC algorithms. The resulting scheme produces an approximate i.i.d. sample from the target distribution and is fully parallelizable, as trajectories can be generated independently. The second class consists of MCMC algorithms targeting the joint law of the whole diffusion path in $[0,T]$, for a suitably chosen horizon $T$. The proposed samplers combine three types of updates. One update simulates the diffusion forward in time according to an OU dynamics, conditional on its initial value. The remaining two update the backward component via Metropolis-type steps: one conditions on the terminal value at time $T$ and the other one does not. In both cases, acceptance probabilities are implemented using Barker-type Bernoulli factory constructions. The proposed methods perform well for targets with multimodality and complex dependence structures, providing a scalable and efficient alternative to the widely used random-walk Metropolis algorithm.

Problem

• Monte Carlo sampling
• unnormalized densities
• reverse diffusion
• multimodality
• complex dependence

Innovation

• reverse diffusion
• score-free sampling
• Radon-Nikodym derivative
• pseudo-marginal MCMC
• Barker Bernoulli factory