Diffusion models provide expressive data-driven priors for Bayesian inverse problems, but many diffusion posterior samplers rely on heuristic guidance approximations that can fail for nonlinear operators and multimodal posteriors. In this work, we develop a stabilized path-space framework for diffusion-based posterior sampling. Starting from a base diffusion process whose terminal marginal represents the prior, we define a likelihood-weighted target measure on trajectories and cast posterior sampling as learning a controlled stochastic process whose path measure matches this target. This formulation connects diffusion posterior sampling to stochastic optimal control while preserving the Bayesian structure needed for uncertainty quantification. We introduce a time reparameterization that makes the path-space control problem well posed by removing the bias induced by the unknown initial value function, without auxiliary training. We then learn the control via a trust-region path-space optimization method with log-variance objectives. The path-space perspective also unifies our learned control approach with existing guidance-based samplers, quantifies the sampling error induced by approximate controls, and yields importance sampling corrections for asymptotically exact posterior expectations. We evaluate the proposed framework on a suite of benchmark inverse problems with analytically characterized or high-quality reference posteriors, enabling principled assessment of sampling accuracy and uncertainty quantification. These experiments provide insight into the behavior of diffusion-based posterior samplers and demonstrate improved accuracy and robustness over leading approaches.

翻译：扩散模型为贝叶斯逆问题提供了表达性强的数据驱动先验，但许多扩散后验采样器依赖启发式引导近似，这可能在非线性算子和多模态后验中失效。本文开发了一种面向扩散后验采样的稳定路径空间框架。从终端边际表示先验的基础扩散过程出发，我们定义了轨迹上的似然加权目标测度，并将后验采样转化为学习一个路径测度与该目标匹配的受控随机过程。该公式将扩散后验采样与随机最优控制联系起来，同时保留了不确定性量化所需的贝叶斯结构。我们引入一种时间重参数化，通过消除未知初值函数引起的偏差，使路径空间控制问题良定，且无需辅助训练。随后通过具有对数方差目标的信赖域路径空间优化方法学习控制。路径空间视角还统一了我们的学习控制方法与现有基于引导的采样器，量化了近似控制引起的采样误差，并给出了用于渐近精确后验期望的重要性采样修正。我们在具有解析表征或高质量参考后验的基准逆问题套件上评估了所提框架，实现了采样精度和不确定性量化的原则性评估。这些实验揭示了扩散后验采样器的行为特征，并展示了相较于主流方法的精度提升和鲁棒性改进。