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93
How to Use Expert Advice
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1997
"... We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worstcase situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the ..."
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Cited by 376 (72 self)
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We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worstcase situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show howthis leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also compare our analysis to the case in which log loss is used instead of the expected number of mistakes.
Strictly Proper Scoring Rules, Prediction, and Estimation
, 2007
"... Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distribution F if he ..."
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Cited by 357 (27 self)
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Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distribution F if he or she issues the probabilistic forecast F, rather than G ̸ = F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the Savage representation. Examples of scoring rules for probabilistic forecasts in the form of predictive densities include the logarithmic, spherical, pseudospherical, and quadratic scores. The continuous ranked probability score applies to probabilistic forecasts that take the form of predictive cumulative distribution functions. It generalizes the absolute error and forms a special case of a new and very general type of score, the energy score. Like many other scoring rules, the energy score admits a kernel representation in terms of negative definite functions, with links to inequalities of Hoeffding type, in both univariate and multivariate settings. Proper scoring rules for quantile and interval forecasts are also discussed. We relate proper scoring rules to Bayes factors and to crossvalidation, and propose a novel form of crossvalidation known as randomfold crossvalidation. A case study on probabilistic weather forecasts in the North American Pacific Northwest illustrates the importance of propriety. We note optimum score approaches to point and quantile
Optimal aggregation of classifiers in statistical learning
 Ann. Statist
, 2004
"... Classification can be considered as nonparametric estimation of sets, where the risk is defined by means of a specific distance between sets associated with misclassification error. It is shown that the rates of convergence of classifiers depend on two parameters: the complexity of the class of cand ..."
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Cited by 225 (7 self)
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Classification can be considered as nonparametric estimation of sets, where the risk is defined by means of a specific distance between sets associated with misclassification error. It is shown that the rates of convergence of classifiers depend on two parameters: the complexity of the class of candidate sets and the margin parameter. The dependence is explicitly given, indicating that optimal fast rates approaching O(n−1) can be attained, where n is the sample size, and that the proposed classifiers have the property of robustness to the margin. The main result of the paper concerns optimal aggregation of classifiers: we suggest a classifier that automatically adapts both to the complexity and to the margin, and attains the optimal fast rates, up to a logarithmic factor. 1. Introduction. Let (Xi,Yi)
InformationTheoretic Determination of Minimax Rates of Convergence
 Ann. Stat
, 1997
"... In this paper, we present some general results determining minimax bounds on statistical risk for density estimation based on certain informationtheoretic considerations. These bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence. ..."
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Cited by 158 (24 self)
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In this paper, we present some general results determining minimax bounds on statistical risk for density estimation based on certain informationtheoretic considerations. These bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence.
Smooth Discrimination Analysis
 Ann. Statist
, 1998
"... Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if di ..."
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Cited by 154 (3 self)
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Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if discriminant analysis is used as a data analytic tool. We discuss optimal rates for estimation of G. 1991 AMS: primary 62G05 , secondary 62G20 Keywords and phrases: discrimination analysis, minimax rates, Bayes risk Short title: Smooth discrimination analysis This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 "Quantifikation und Simulation okonomischer Prozesse", HumboldtUniversitat zu Berlin 1 Introduction Assume that one observes two independent samples X = (X 1 ; : : : ; X n ) and Y = (Y 1 ; : : : ; Ym ) of IR d valued i.i.d. observations with densities f or g, respectively. The densities f and g are unknown. An additional random variabl...
Ideal denoising in an orthonormal basis chosen from a library of bases
 Comptes Rendus Acad. Sci., Ser. I
, 1994
"... of bases ..."
Convergence rates of posterior distributions
 Ann. Statist
, 2000
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, D ..."
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Cited by 106 (14 self)
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We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, Dirichlet processes and interval censoring. 1. Introduction. Suppose
Theory of classification: A survey of some recent advances
, 2005
"... The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results. ..."
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Cited by 93 (3 self)
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The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results.
Calibrated Probabilistic Forecasting Using Ensemble Model Output Statistics and Minimum CRPS Estimation
 MONTHLY WEATHER REVIEW VOLUME
, 2005
"... Ensemble prediction systems typically show positive spreaderror correlation, but they are subject to forecast bias and dispersion errors, and are therefore uncalibrated. This work proposes the use of ensemble model output statistics (EMOS), an easytoimplement postprocessing technique that address ..."
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Cited by 78 (14 self)
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Ensemble prediction systems typically show positive spreaderror correlation, but they are subject to forecast bias and dispersion errors, and are therefore uncalibrated. This work proposes the use of ensemble model output statistics (EMOS), an easytoimplement postprocessing technique that addresses both forecast bias and underdispersion and takes into account the spreadskill relationship. The technique is based on multiple linear regression and is akin to the superensemble approach that has traditionally been used for deterministicstyle forecasts. The EMOS technique yields probabilistic forecasts that take the form of Gaussian predictive probability density functions (PDFs) for continuous weather variables and can be applied to gridded model output. The EMOS predictive mean is a biascorrected weighted average of the ensemble member forecasts, with coefficients that can be interpreted in terms of the relative contributions of the member models to the ensemble, and provides a highly competitive deterministicstyle forecast. The EMOS predictive variance is a linear function of the ensemble variance. For fitting the EMOS coefficients, the method of minimum continuous ranked probability score (CRPS) estimation is introduced. This technique finds the coefficient values that optimize the CRPS for the training data. The EMOS technique was applied to 48h forecasts of sea level pressure and surface temperature over the North American Pacific Northwest in spring 2000, using the University of Washington mesoscale ensemble. When compared to the biascorrected ensemble, deterministicstyle EMOS forecasts of sea level pressure had rootmeansquare error 9 % less and mean absolute error 7 % less. The EMOS predictive PDFs were sharp, and much better calibrated than the raw ensemble or the biascorrected ensemble.