損失関数
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2023/06/05 21:06 UTC 版)
決定則
決定則（decision rules）は、最適化基準を使用して選択を行うものである。よく使われる基準として次のようなものがある。
- ミニマックス（minimax）最悪の損失が最も少ない決定則を選ぶ。つまり最悪の場合の損失（最大可能損失）を最小限に抑える。
- 不変性（invariance）：不変性要件を満たす決定則を選択する。
- 平均損失が最も少ない（つまり損失関数の期待値を最小化する）決定則を選ぶ。
![{\displaystyle {\underset {\delta }{\operatorname {arg\,min} }}\operatorname {E} _{\theta \in \Theta }[R(\theta ,\delta )]={\underset {\delta }{\operatorname {arg\,min} }}\ \int _{\theta \in \Theta }R(\theta ,\delta )\,p(\theta )\,d\theta .}](https://weblio.hs.llnwd.net/e7/img/dict/wkpja/https://wikimedia.org/api/rest_v1/media/math/render/svg/118ed61f14f1d53953242831140397d033eb052d)
- ^ a b Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome H. (2001). The Elements of Statistical Learning. Springer. p. 18. ISBN 0-387-95284-5
- ^ Wald, A. (1950). Statistical Decision Functions. Wiley
- ^ Cramér, H. (1930). On the mathematical theory of risk
- ^ Frisch, Ragnar (1969). “From utopian theory to practical applications: the case of econometrics”. The Nobel Prize–Prize Lecture 2021年2月15日閲覧。
- ^ Tangian, Andranik; Gruber, Josef (1997). Constructing Scalar-Valued Objective Functions. Proceedings of the Third International Conference on Econometric Decision Models: Constructing Scalar-Valued Objective Functions, University of Hagen, held in Katholische Akademie Schwerte September 5–8, 1995. Lecture Notes in Economics and Mathematical Systems. 453. Berlin: Springer. doi:10.1007/978-3-642-48773-6. ISBN 978-3-540-63061-6
- ^ Tangian, Andranik; Gruber, Josef (2002). Constructing and Applying Objective Functions. Proceedings of the Fourth International Conference on Econometric Decision Models Constructing and Applying Objective Functions, University of Hagen, held in Haus Nordhelle, August, 28 — 31, 2000. Lecture Notes in Economics and Mathematical Systems. 510. Berlin: Springer. doi:10.1007/978-3-642-56038-5. ISBN 978-3-540-42669-1
- ^ Tangian, Andranik (2002). “Constructing a quasi-concave quadratic objective function from interviewing a decision maker”. European Journal of Operational Research 141 (3): 608–640. doi:10.1016/S0377-2217(01)00185-0.
- ^ Tangian, Andranik (2004). “A model for ordinally constructing additive objective functions”. European Journal of Operational Research 159 (2): 476–512. doi:10.1016/S0377-2217(03)00413-2.
- ^ Tangian, Andranik (2004). “Redistribution of university budgets with respect to the status quo”. European Journal of Operational Research 157 (2): 409–428. doi:10.1016/S0377-2217(03)00271-6.
- ^ Tangian, Andranik (2008). “Multi-criteria optimization of regional employment policy: A simulation analysis for Germany”. Review of Urban and Regional Development 20 (2): 103–122. doi:10.1111/j.1467-940X.2008.00144.x.
- ^ Pfanzagl, J. (1994). Parametric Statistical Theory. Berlin: Walter de Gruyter. ISBN 978-3-11-013863-4
- ^ Detailed information on mathematical principles of the loss function choice is given in Chapter 2 of the book Klebanov, B.; Rachev, Svetlozat T.; Fabozzi, Frank J. (2009). Robust and Non-Robust Models in Statistics. New York: Nova Scientific Publishers, Inc. (and references there).
- ^ Deming, W. Edwards (2000). Out of the Crisis. The MIT Press. ISBN 9780262541152
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