Diffusion-Based Posterior Sampling Method

• Diffusion-based posterior sampling is a method that uses score-based generative diffusion models to sample from complex Bayesian posteriors.
• It leverages a forward noising process and a trained neural network score estimator to reverse the stochastic process and generate data.
• Recent advances enhance computational efficiency and model flexibility, addressing challenges in sequential, nonlinear, and spatiotemporal inverse problems.

A diffusion-based posterior sampling method employs score-based generative diffusion models to sample from posteriors conditioning on observed measurements within Bayesian inverse problems. In the context of high-dimensional imaging or video, these methods leverage deep models trained to approximate the time-dependent score function of the data prior, enabling sampling from complex posterior distributions through learned reverse stochastic processes. Recent advances address both computational efficiency and model flexibility for challenging real-world settings, extending to sequential, nonlinear, and spatiotemporal inverse problems. This article provides a comprehensive technical survey of diffusion-based posterior sampling approaches, with emphasis on methods, acceleration strategies, empirical properties, and application-specific considerations (Stevens et al., 9 Sep 2024, Chung et al., 2022, Zhou et al., 15 Nov 2024, Cao et al., 30 Aug 2024, Li et al., 2023, Li et al., 13 Mar 2025).

1. Mathematical and Algorithmic Foundations

Given a forward model $y = A x_0 + \eta$, with $x_0\in\mathbb{R}^n$ the latent image or frame, $y\in\mathbb{R}^m$ the observation, $A$ a (possibly time-varying or nonlinear) operator, and $\eta\sim\mathcal{N}(0,\sigma^2 I)$, the goal is to sample from the posterior $p(x_0|y) \propto p(y|x_0)p(x_0)$. The prior $p(x_0)$ is modeled implicitly via a pretrained diffusion model, which defines a forward noising SDE: $$dx = -\frac{1}{2}\beta(t)x\,dt + \sqrt{\beta(t)}\,dw,$$ or its discrete form

$$q(x_i|x_{i-1}) = \mathcal{N}(\sqrt{1-\beta_i}x_{i-1}, \beta_i I).$$

Reverse-time sampling is achieved using a trained neural network score estimator $s_\theta(x,t)\approx \nabla_x \log p_t(x)$ in the corresponding reverse process: