This paper presents a novel method for training diffusion models from incomplete and noisy observations using the expectation-maximization (EM) algorithm. The method addresses the challenge of learning proper diffusion models, which are crucial for downstream tasks in Bayesian inference. Unlike previous works, the proposed method ensures that the learned prior is a valid probability distribution, making it suitable for empirical Bayes problems. The authors introduce an improved posterior sampling scheme, named Moment Matching Posterior Sampling (MMPS), which provides more accurate and stable samples compared to existing methods. The effectiveness of the proposed method is demonstrated through experiments on low-dimensional manifolds, corrupted CIFAR-10 images, and accelerated MRI data. The results show that the method can recover high-quality samples that are consistent with the observations, outperforming previous approaches in terms of sample quality and downstream task performance.This paper presents a novel method for training diffusion models from incomplete and noisy observations using the expectation-maximization (EM) algorithm. The method addresses the challenge of learning proper diffusion models, which are crucial for downstream tasks in Bayesian inference. Unlike previous works, the proposed method ensures that the learned prior is a valid probability distribution, making it suitable for empirical Bayes problems. The authors introduce an improved posterior sampling scheme, named Moment Matching Posterior Sampling (MMPS), which provides more accurate and stable samples compared to existing methods. The effectiveness of the proposed method is demonstrated through experiments on low-dimensional manifolds, corrupted CIFAR-10 images, and accelerated MRI data. The results show that the method can recover high-quality samples that are consistent with the observations, outperforming previous approaches in terms of sample quality and downstream task performance.