Differential geometrical approach to Bayesian prediction theory to classical and quantum correlated systems

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We obtain a useful parametrization of superharmonic priors in AR models, which are often used in time series analysis. It is shown to be useful not only in theoretical development but also in numerical analysis. We show that there is no alpha parallel prior except for the Jeffreys prior in the ARMA model by using differential geometry. We give a general definition of noninformative prior in a statistical model of wave functions and prove a minimax theorem with respect to this prior. We also derive the optimal estimation of wave functions and give an explicit example where our estimator has a uniformly better performance than that based on MLE.