  This paper considers the objective comparison of stochastic models to solve
inverse problems, more specifically image restoration. Most often, model
comparison is addressed in a supervised manner, that can be time-consuming and
partly arbitrary. Here we adopt an unsupervised Bayesian approach and
objectively compare the models based on their posterior probabilities, directly
from the data without ground truth available. The probabilities depend on the
marginal likelihood or "evidence" of the models and we resort to the Chib
approach including a Gibbs sampler. We focus on the family of Gaussian models
with circulant covariances and unknown hyperparameters, and compare different
types of covariance matrices for the image and noise.
