  In recent years, denoising problems have become intertwined with the
development of deep generative models. In particular, diffusion models are
trained like denoisers, and the distribution they model coincide with denoising
priors in the Bayesian picture. However, denoising through diffusion-based
posterior sampling requires the noise level and covariance to be known,
preventing blind denoising. We overcome this limitation by introducing Gibbs
Diffusion (GDiff), a general methodology addressing posterior sampling of both
the signal and the noise parameters. Assuming arbitrary parametric Gaussian
noise, we develop a Gibbs algorithm that alternates sampling steps from a
conditional diffusion model trained to map the signal prior to the family of
noise distributions, and a Monte Carlo sampler to infer the noise parameters.
Our theoretical analysis highlights potential pitfalls, guides diagnostic
usage, and quantifies errors in the Gibbs stationary distribution caused by the
diffusion model. We showcase our method for 1) blind denoising of natural
images involving colored noises with unknown amplitude and spectral index, and
2) a cosmology problem, namely the analysis of cosmic microwave background
data, where Bayesian inference of "noise" parameters means constraining models
of the evolution of the Universe.
