In statistical classification the **Bayes classifier** minimizes the probability of misclassification.^{[1]}

## Definition

Suppose a pair takes values in , where is the class label of . This means that the conditional distribution of *X*, given that the label *Y* takes the value *r* is given by

- for

where "" means "is distributed as", and where denotes a probability distribution.

A classifier is a rule that assigns to an observation *X*=*x* a guess or estimate of what the unobserved label *Y*=*r* actually was. In theoretical terms, a classifier is a measurable function , with the interpretation that *C* classifies the point *x* to the class *C*(*x*). The probability of misclassification, or risk, of a classifier *C* is defined as

The Bayes classifier is

In practice, as in most of statistics, the difficulties and subtleties are associated with modeling the probability distributions effectively—in this case, . The Bayes classifier is a useful benchmark in statistical classification.

The excess risk of a general classifier (possibly depending on some training data) is defined as
Thus this non-negative quantity is important for assessing the performance of different classification techniques. A classifier is said to be consistent if the excess risk converges to zero as the size of the training data set tends to infinity.^{[citation needed]}

## See also

## References

**^**Devroye, L.; Gyorfi, L. & Lugosi, G. (1996).*A probabilistic theory of pattern recognition*. Springer. ISBN 0-3879-4618-7.