Information geometry of Bayesian statistics

A survey of geometry of Bayesian statistics is given. From the viewpoint of differential geometry, a prior distribution in Bayesian statistics is regarded as a volume element on a statistical model. In this paper, properties of Bayesian estimators are studied by applying equiaffine structures of statistical manifolds. In addition, geometry of anomalous statistics is also studied. Deformed expectations and deformed independeces are important in anomalous statistics. After summarizing geometry of such deformed structues, a generalization of maximum likelihood method is given. A suitable weight on a parameter space is important in Bayesian statistics, whereas a suitable weight on a sample space is important in anomalous statistics.