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March, 1974 Optimal Predictive Linear Discriminants
Peter Enis, Seymour Geisser
Ann. Statist. 2(2): 403-410 (March, 1974). DOI: 10.1214/aos/1176342677


When classifying an observation $\mathbf{z}$ which has arisen (with known prior probabilities) from one of two $p$-variate nonsingular normal populations with known parameters, the discriminant, say $U$, which minimizes the total probability of misclassification is based on the logarithm of the ratio of the densities of the two populations. When the parameters are unknown, the "classical" procedure has been to substitute sample estimates for the unknown parameters in $U$ and use the resulting sample discriminant, say $V$, as the basis for classifying future observations. This procedure need not enjoy the property of minimizing the probability of misclassification and has been justified, from the classical point of view, almost entirely on the grounds that it seems intuitively reasonable. When the covariance matrices of the two normal populations are equal, $U$ is a linear function of the observation vector $\mathbf{z}$. The fact that $U$ minimizes the probability of misclassification does not imply that $V$ will. Further, although $U$ is linear, the sample discriminant which minimizes the probability of misclassification will, in general, not be linear. Here, using the Bayesian notion of a predictive distribution, we obtain from amongst the class of linear sample discriminants that one which minimizes the predictive probability of misclassification.


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Peter Enis. Seymour Geisser. "Optimal Predictive Linear Discriminants." Ann. Statist. 2 (2) 403 - 410, March, 1974.


Published: March, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0449.62043
MathSciNet: MR395058
Digital Object Identifier: 10.1214/aos/1176342677

Primary: 62H30
Secondary: 62F15

Keywords: ‎classification‎ , discrimination , linear discriminants , predictive discriminants

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 2 • March, 1974
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