The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ that specify the prior and then selecting the one with the highest value. Diffusion models are the state-of-the-art approach to solving inverse problems with a data-driven prior; however, directly computing the model evidence with respect to a diffusion prior is intractable. Furthermore, most existing model evidence estimators require either many pointwise evaluations of the unnormalized prior density or an accurate clean prior score. We propose DiME, an estimator of the model evidence of a diffusion prior by integrating over the time-marginals of posterior sampling methods. Our method leverages the large amount of intermediate samples naturally obtained during the reverse diffusion sampling process to obtain an accurate estimation of the model evidence using only a handful of posterior samples (e.g., 20). We also demonstrate how to implement our estimator in tandem with recent diffusion posterior sampling methods. Empirically, our estimator matches the model evidence when it can be computed analytically, and it is able to both select the correct diffusion model prior and diagnose prior misfit under different highly ill-conditioned, non-linear inverse problems, including a real-world black hole imaging problem.

翻译：先验选择是解决病态成像逆问题的核心，因此必须选择与测量值$y$一致的先验以避免严重偏差。在贝叶斯逆问题中，可通过评估不同指定先验的模型$M$下的模型证据$p(y \mid M)$，并选择值最高的模型来实现这一目标。扩散模型是当前利用数据驱动先验解决逆问题的最先进方法，但直接计算扩散先验下的模型证据是不可行的。此外，现有的大多数模型证据估计器要么需要对未归一化先验密度进行大量逐点评估，要么需要准确的干净先验评分。我们提出DiME方法，通过整合后验采样方法的时间边际分布来估计扩散先验的模型证据。该方法利用逆向扩散采样过程中自然获得的大量中间样本，仅需少量后验样本（例如20个）即可精确估计模型证据。我们还展示了如何将所提估计器与最新扩散后验采样方法协同实现。实验表明，在可解析计算模型证据时，我们的估计器与之匹配；在高度病态的非线性逆问题（包括真实黑洞成像问题）中，该方法既能正确选择扩散模型先验，又能诊断先验失配。