A novel approach named DiEM has been developed to train diffusion models using incomplete and noisy datasets, addressing a key limitation in the field of Bayesian inverse problems. This method leverages the expectation-maximization algorithm to effectively learn diffusion priors from observational data, thereby reducing the reliance on large volumes of clean data typically required for such models. By tackling the challenge of data quality and availability, DiEM represents a significant advancement in machine learning techniques related to diffusion processes. The approach’s foundation in expectation-maximization allows it to iteratively refine model parameters despite data imperfections. This innovation is particularly relevant for applications where acquiring pristine datasets is impractical or costly. The introduction of DiEM has been positively received as a meaningful contribution to the broader AI research community focused on probabilistic modeling and inverse problem-solving. Its development aligns with ongoing efforts to enhance model robustness and applicability in real-world scenarios characterized by noisy and incomplete information.