  Prior design is one of the most important problems in both statistics and
machine learning. The cross validation (CV) and the widely applicable
information criterion (WAIC) are predictive measures of the Bayesian
estimation, however, it has been difficult to apply them to find the optimal
prior because their mathematical properties in prior evaluation have been
unknown and the region of the hyperparameters is too wide to be examined. In
this paper, we derive a new formula by which the theoretical relation among CV,
WAIC, and the generalization loss is clarified and the optimal hyperparameter
can be directly found.
  By the formula, three facts are clarified about predictive prior design.
Firstly, CV and WAIC have the same second order asymptotic expansion, hence
they are asymptotically equivalent to each other as the optimizer of the
hyperparameter. Secondly, the hyperparameter which minimizes CV or WAIC makes
the average generalization loss to be minimized asymptotically but does not the
random generalization loss. And lastly, by using the mathematical relation
between priors, the variances of the optimized hyperparameters by CV and WAIC
are made smaller with small computational costs. Also we show that the
optimized hyperparameter by DIC or the marginal likelihood does not minimize
the average or random generalization loss in general.
