ABSTRACT
This chapter explains that many models and priors can be made into standard forms by using the algebraic geometric transform. Then the posterior distribution is represented as a finite mixture of locally standard forms. In Bayesian estimation, the set of parameters can be understood as a union of local parameter sets. Resolution theorem in algebraic geometry makes an arbitrary statistical model a locally standard form. The chapter examines the general theory of Bayesian statistics and describes the generalization losses of the maximum likelihood and a posteriori methods. In Bayesian statistics, the posterior distribution can be decomposed as a sum of local distributions. Even if the posterior distribution cannot be approximated by any normal distribution, there exists division of parameter set such that the average log density ratio function can be normal crossing in each local parameter set. The resolution theorem in algebraic geometry is the mathematical base for statistical analysis of general Bayesian statistics.
