Abstract
This paper presents direct settings and rigorous solutions of Statistical Inference problems. It shows that rigorous solutions require solving ill-posed Fredholm integral equations of the first kind in the situation where not only the right-hand side of the equation is an approximation, but the operator in the equation is also defined approximately. Using Stefanuyk-Vapnik theory for solving such operator equations, constructive methods of empirical inference are introduced. These methods are based on a new concept called \(V\)-matrix. This matrix captures geometric properties of the observation data that are ignored by classical statistical methods.
This material is based upon work partially supported by AFRL and DARPA under contract FA8750-14-C-0008. Any opinions, findings and or conclusions in this material are those of the authors and do not necessarily reflect the views of AFRL and DARPA.
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References
Cover, T., Thomas, J.: Elements of Information Theory. Wiley, New York (1991)
Kawahara Y., Sugiyama, M.: Change-point detection in time-series data by direct density-ratio estimation. In: Proceedings of the 2009 SIAM International Conference on Data Mining, pp. 389–400 (2009)
Kimeldorf, G., Wahba, G.: Some Results on Tchebycheffian Spline Functions. Journal of Mathematical Analysis and Applications. 33, 82–95 (1971)
Kimeldorf, G., Wahba, G.: A Correspondence between Bayesian Estimation on Stochastic Processes and Smoothing by Splines. Annals of Mathematical Statistics 41(2), 495–502 (1970)
Massart, P.: The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality. Annals of Probability 18(3), 1269–1283 (1990)
Osuna, E., Girosi, F.: Reducing the run-time complexity in support vector machines. In: Advances in Kernel Methods, pp. 271–283. MIT Press, Cambridge (1999)
Saunders, C., Gammerman, A., Vovk, A.: Ridge regression learning algorithm in dual variables. In: 15th International Conference on Machine Learning, pp. 515–521 (1998)
Schölkopf, B., Herbrich, R., Smola, A.J.: A Generalized Representer Theorem. In: Helmbold, D.P., Williamson, B. (eds.) COLT 2001 and EuroCOLT 2001. LNCS (LNAI), vol. 2111, p. 416. Springer, Heidelberg (2001)
Stefanyuk, A.: Estimation of the Likelihood Ratio Function in the “Disorder” Problem of Random Processes. Automation and Remote Control 9, 53–59 (1986)
Steinwart, I., Scovel, C. When do support machines learn fast? In: 16th International Symposium on Mathematical Theory of Networks and Systems (2004)
Sugiyama, M., Suzuki, T., Kanamori, T.: Density Ratio Estimation in Machine Learning. Cambridge University Press (2011)
Suykens, J., Vandewalle, J.: Least Squares Support Vector Machine Classifiers. Neural Processing Letters 9(3), 293–300 (1999)
Tikhonov, A., Arsenin, V.: Solution of Ill-Posed Problems. Winston & Sons (1977)
Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1995)
Vapnik, V.: Statistical Learning Theory. John Wiley & Sons, New York (1998)
Vapnik, V., Braga, I., Izmailov, R.: A constructive setting for the problem of density ratio estimation. In: SIAM International Conference on Data Mining, pp. 434–442 (2014)
Vapnik, V., Chervonenkis, A.: Theory of Pattern Recognition (in Russian). Nauka, Moscow (1974). German translation: Wapnik W., Tscherwonenkis, A.: Theorie des Zeichenerkennung. Akademie-Verlag, Berlin (1974)
Vapnik, V., Izmailov, R.: Learning with Intelligent Teacher: Similarity Control and Knowledge Transfer. In: Statistical Learning and Data Sciences. SLDS 2015, LNCS (LNAI), vol. 9047, pp. 3–32. Springer-Verlag, London (2015)
Vapnik, V., Stefanyuk, A.: Nonparametric Methods for Estimating Probability Densities. Automation and Remote Control 8, 38–52 (1978)
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Vapnik, V., Izmailov, R. (2015). Statistical Inference Problems and Their Rigorous Solutions. In: Gammerman, A., Vovk, V., Papadopoulos, H. (eds) Statistical Learning and Data Sciences. SLDS 2015. Lecture Notes in Computer Science(), vol 9047. Springer, Cham. https://doi.org/10.1007/978-3-319-17091-6_2
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DOI: https://doi.org/10.1007/978-3-319-17091-6_2
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