Abstract #300903

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JSM 2003 Abstract #300903
Activity Number: 25
Type: Contributed
Date/Time: Sunday, August 3, 2003 : 2:00 PM to 3:50 PM
Sponsor: IMS
Abstract - #300903
Title: The Benefits of Assuming Independence when There Are Many More Variables than Observations: A Theory for Fisher's Linear Discriminant
Author(s): Elizaveta Levina*+ and Peter J. Bickel
Companies: University of Michigan and University of California, Berkeley
Address: Dept. of Statistics, Ann Arbor, MI, 48109-1092,
Keywords: naive Bayes ; Fisher's linear discriminant ; classification ; high-dimensional data ; Gaussian colored noise
Abstract:

It is well known in machine learning practice that assuming independent covariates in classification is highly beneficial when there are more covariates than observations, a phenomenon we have encountered in the complicated context of texture classification. Here we consider the issue in the classical context of discriminating between two normal populations, and prove that the "naïve Bayes'' classifier greatly outperforms the Fisher linear discriminant rule under broad conditions when the number of variables grows faster than the number of observations. We also introduce a class of rules spanning the range between independence and arbitrary dependence. These rules are shown to achieve Bayes consistency for the Gaussian "colored noise'' model and to adapt to a spectrum of convergence rates, which we conjecture to be minimax.

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