Theoretical study of hyperparameter estimation by maximization of marginal likelihood in image restoration by means of cluster variation method

From the perspective of maximum likelihood estimation, the hyperparameters that appear in the statistical methods for image restoration are determined by maximizing the quantity referred to as the marginal likelihood. Rigorously calculating the marginal likelihood for the Markov random field model is extremely difficult, and approximate analysis techniques must be introduced. A well-known conventional approximate analysis technique is mean field approximation. In this paper, we propose a novel approximate analysis technique that uses the cluster variation method instead of mean field approximation to more accurately estimate hyperparameters than estimation using the marginal likelihood. The cluster variation method is a typical approximate analysis technique in statistical mechanics obtained by extending mean field approximation and can obtain some statistical quantities of the Markov random field model with high accuracy. However, the cluster variation method has not been applied to hyperparameter estimation employing the marginal likelihood. Compared to the results of earlier mean field approximation often used as the marginal likelihood approximate analysis technique, numerical experiments show that hyperparameter estimation using the cluster variation method produces better restorations particularly of images that closely reflect the prior probability distribution.