  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.
