Accuracy of the Bethe approximation for hyperparameter estimation in probabilistic image processing

Kazuyuki Tanaka, Hayaru Shouno, Masato Okada, D. M. Titterington

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)


We investigate the accuracy of statistical-mechanical approximations for the estimation of hyperparameters from observable data in probabilistic image processing, which is based on Bayesian statistics and maximum likelihood estimation. Hyperparameters in statistical science correspond to interactions or external fields in the statistical-mechanics context. In this paper, hyperparameters in the probabilistic model are determined so as to maximize a marginal likelihood. A practical algorithm is described for grey-level image restoration based on a Gaussian graphical model and the Bethe approximation. The algorithm corresponds to loopy belief propagation in artificial intelligence. We examine the accuracy of hyperparameter estimation when we use the Bethe approximation. It is well known that a practical algorithm for probabilistic image processing can be prescribed analytically when a Gaussian graphical model is adopted as a prior probabilistic model in Bayes' formula. We are therefore able to compare, in a numerical study, results obtained through mean-field-type approximations with those based on exact calculation.

Original languageEnglish
Pages (from-to)8675-8695
Number of pages21
JournalJournal of Physics A: Mathematical and General
Issue number36
Publication statusPublished - 2004 Sept 10


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