Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, our method leads to proper diffusion models, which is crucial for downstream tasks. As part of our method, we propose and motivate a new posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our method.

翻译：扩散模型最近被证明是解决贝叶斯逆问题的卓越先验模型。然而，训练这些模型通常需要大量干净数据，这在某些场景下可能难以实现。本研究提出了一种基于期望最大化算法的新方法，仅通过不完整且含噪声的观测数据来训练扩散模型。与先前工作不同，我们的方法能够生成规范的扩散模型，这对下游任务至关重要。作为方法的一部分，我们提出并论证了一种针对无条件扩散模型的新后验采样方案。我们通过实证证据验证了该方法的有效性。