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Worst-case bounds for the logarithmic loss of predictors




We investigate on-line prediction of individual sequences. Given a class of predictors, the goal is to predict as well as the best predictor in the class, where the loss is measured by the self information (logarithmic) loss function. The excess loss (regret) is closely related to the redundancy of the associated lossless universal code. Using Shtarkov's theorem and tools from empirical process theory, we prove a general upper bound on the best possible (minimax) regret. The bound depends on certain metric properties of the class of predictors. We apply the bound to both parametric and nonparametric classes of predictors. Finally, we point out a suboptimal behavior of the popular Bayesian weighted average algorithm.

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  • Nicolò Cesa Bianchi & Gábor Lugosi, 1999. "Worst-case bounds for the logarithmic loss of predictors," Economics Working Papers 418, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:418

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    References listed on IDEAS

    1. Becker, Gary S, 1973. "A Theory of Marriage: Part I," Journal of Political Economy, University of Chicago Press, vol. 81(4), pages 813-846, July-Aug..
    2. Levy, Frank & Murnane, Richard J, 1992. "U.S. Earnings Levels and Earnings Inequality: A Review of Recent Trends and Proposed Explanations," Journal of Economic Literature, American Economic Association, vol. 30(3), pages 1333-1381, September.
    3. Juhn, Chinhui & Murphy, Kevin M & Pierce, Brooks, 1993. "Wage Inequality and the Rise in Returns to Skill," Journal of Political Economy, University of Chicago Press, vol. 101(3), pages 410-442, June.
    4. Epple, Dennis & Romano, Richard E, 1998. "Competition between Private and Public Schools, Vouchers, and Peer-Group Effects," American Economic Review, American Economic Association, vol. 88(1), pages 33-62, March.
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    Cited by:

    1. Alessio Sancetta, 2010. "Bootstrap model selection for possibly dependent and heterogeneous data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(3), pages 515-546, June.
    2. Yuan Lo-Hua & Liu Anthony & Yeh Alec & Franks Alex & Wang Sherrie & Illushin Dmitri & Bornn Luke & Kaufman Aaron & Reece Andrew & Bull Peter, 2015. "A mixture-of-modelers approach to forecasting NCAA tournament outcomes," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 11(1), pages 13-27, March.

    More about this item


    Universal prediction; universal coding; empirical processes; on-line learning; metric entropy;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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