Information Geometry, Bayesian Inference, Ideal Estimates, and Error Decomposition

Huaiyu Zhu and Richard Rohwer

98-06-045

In statistics it is necessary to study the relation among many probability distributions. Information geometry elucidates the geometric structure on the space of all distributions. When combined with Bayesian decision theory, it leads to the new concept of "ideal estimates." They uniquely exist in the space of finite measures, and are generally sufficient statistic. The optimal estimate on any model is given by projecting the ideal estimate onto that model. An error decomposition theorem splits the error of an estimate into the sum of statistical error and approximation error. They can be expanded to yield higher-order asymptotics. Furthermore, the ideal estimates under certain uniform priors, invariantly defined in information geometry, corresponds to various optimal non-Bayesian estimates, such as the MLE.