The Nishimori line and Bayesian Statistics

``Nishimori line'' is a line or hypersurface in the parameter space of systems with quenched disorder, where simple expressions of the averages of physical quantities over the quenched random variables are obtained. It has been playing an important role in the theoretical studies of the random frustrated systems since its discovery around 1980. In this paper, a novel interpretation of the Nishimori line from the viewpoint of statistical information processing is presented. Our main aim is the reconstruction of the whole theory of the Nishimori line from the viewpoint of Bayesian statistics, or, almost equivalently, from the viewpoint of the theory of error-correcting codes. As a byproduct of our interpretation, counterparts of the Nishimori line in models without gauge invariance are given. We also discussed the issues on the ``finite temperature decoding'' of error-correcting codes in connection with our theme and clarify the role of gauge invariance in this topic.