(Postal Mail)
Sumio Watanabe, Ph.D. Professor Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Mailbox: G519 4259 Nagatsuta, Midoriku, Yokohama, 2268502 Japan. (Email) swatanab (AT) dis . titech . ac . jp (Remark) Old address, swatanab (AT) pi . titech . ac . jp, is not available now. Japanese Homepage DBLP: Computer Science Bibliography Paper Information Return to Watanabe Lab. 
In 1998, we found a bridge between algebraic geometry and learning theory. 
Beyond Laplace and Fisher
WAIC and WBIC 

Sumio Watanabe, Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, 2009. New statistical theory is established that holds even for nonregular models such as a normal mixture, a neural network, and hidden Markov models. The resolution theorem in algebraic geometry transforms the likelihood function to a new standard form in statistics. The asymptotic behavior of the log likelihood ratio function is given by the limit empirical process on algebraic variety. This theory contains regular statistical theory as a very special part. We can make generalized concepts of AIC and BIC, even if a true distribution is unrealizable by or singular for a statistical model. In fact, WAIC and WBIC are derived. It is very easy to apply them to practical applications. Both WAIC and WBIC are based on the completely new statistical theory. Neither positive definiteness of Fisher information matrix, asymptotic normality of MLE, nor Laplace approximation is necessary in our new theory. Thus, our theory holds for wide range of statistical models. 
Let's compare WAIC with DIC. Let's compare CV with WAIC. Let's compare PSISCV with WAIC. Let's try WBIC 