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
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.