## Bernoulli

- Bernoulli
- Volume 10, Number 6 (2004), 989-1010.

### Some theory for Fisher's linear discriminant function, `naive Bayes', and some alternatives when there are many more variables than observations

Peter J. Bickel and Elizaveta Levina

#### Abstract

We show that the `naive Bayes' classifier which assumes independent covariates greatly outperforms the Fisher linear discriminant rule under broad conditions when the number of variables grows faster than the number of observations, in the classical problem of discriminating between two normal populations. 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 `coloured noise' model and to adapt to a spectrum of convergence rates, which we conjecture to be minimax.

#### Article information

**Source**

Bernoulli, Volume 10, Number 6 (2004), 989-1010.

**Dates**

First available in Project Euclid: 21 January 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1106314847

**Digital Object Identifier**

doi:10.3150/bj/1106314847

**Mathematical Reviews number (MathSciNet)**

MR2108040

**Zentralblatt MATH identifier**

1064.62073

**Keywords**

Fisher's linear discriminant Gaussian coloured noise minimax regret naive Bayes

#### Citation

Bickel, Peter J.; Levina, Elizaveta. Some theory for Fisher's linear discriminant function, `naive Bayes', and some alternatives when there are many more variables than observations. Bernoulli 10 (2004), no. 6, 989--1010. doi:10.3150/bj/1106314847. https://projecteuclid.org/euclid.bj/1106314847