Achievability of Asymptotic Minimax Regret in Online and Batch Prediction

The normalized maximum likelihood model achieves the minimax coding (log-loss) regret for data of fixed sample size n. However, it is a batch strategy, i.e., it requires that n be known in advance. Furthermore, it is computationally infeasible for most statistical models, and several computationally feasible alternative strategies have been devised. We characterize the achievability of asymptotic minimaxity by batch strategies (i.e., strategies that depend on n) as well as online strategies (i.e., strategies independent of n). On one hand, we conjecture that for a large class of models, no online strategy can be asymptotically minimax. We prove that this holds under a slightly stronger definition of asymptotic minimaxity. Our numerical experiments support the conjecture about non-achievability by so called last-step minimax algorithms, which are independent of n. On the other hand, we show that in the multinomial model, a Bayes mixture defined by the conjugate Dirichlet prior with a simple dependency on n achieves asymptotic minimaxity for all sequences, thus providing a simpler asymptotic minimax strategy compared to earlier work by Xie and Barron. The numerical results also demonstrate superior finite-sample behavior by a number of novel batch and online algorithms.Peer reviewe

Bayesian properties of normalized maximum likelihood and its fast computation

Peer reviewe

Structure of Meridional Circulation in Southern Ocean captured by long term Mooring

The Tenth Symposium on Polar Science/Ordinary sessions: [OM] Polar Meteorology and Glaciology, Wed. 4 Dec. / Entrance Hall (1st floor) , National Institute of Polar Researc

確率測度の最適化と レ ート 歪み関数

Open House, ISM in Tachikawa, 2013.6.14統計数理研究所オープンハウス（立川）、H25.6.14ポスター発