Higher Order Equivalence of Bayes Cross Validation and WAIC

  • Sumio WatanabeEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 252)


It was proved in the previous paper (Watanabe, J Mach Learn Res, 11:3571–3591, (2010), [16]) that Bayes cross validation is asymptotically equivalent to the widely applicable information criterion (WAIC), even if the posterior distribution can not be approximated by any normal distribution. In the present paper, we prove that they are equivalent to each other according to the second order asymptotics, if the posterior distribution can be approximated by some normal distribution. Based on this equivalence, it is also shown that the Bayes cross validation and WAIC are asymptotically equivalent as functions of a hyperparameter of a prior.


Cross validation WAIC Asymptotic equivalence Second order 



This research was partially supported by the Ministry of Education, Science, Sports and Culture in Japan, Grant-in-Aid for Scientific Research 23500172, 25120013, and 15K00331.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan

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