---
title: Diffusion-Based Posterior Sampling Method
url: https://www.emergentmind.com/topics/diffusion-based-posterior-sampling-method
type: topic
---

# Diffusion-Based Posterior Sampling Method

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 [2409.05399][2209.14687][2411.09850][2408.17333][2312.01464][2503.10237].

## 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:

Source: https://www.emergentmind.com/topics/diffusion-based-posterior-sampling-method