In statistical classification, the Bayes classifier minimizes the probability of misclassification.
Suppose a pair takes values in , where is the class label of . Assume 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.
Considering the components of to be mutually independent, we get the naive bayes classifier, where
Proof of Optimality
Proof that the Bayes classifier is optimal and Bayes error rate is minimal proceeds as follows.
Define the variables: Risk , Bayes risk , all possible classes to which the points can be classified . Let the posterior probability of a point belonging to class 1 be . Define the classifier as
Then we have the following results:
(a) , i.e. is a Bayes classifier,
(b) For any classifier , the excess risk satisfies
Proof of (a): For any classifier , we have
(due to Fubini's theorem)
Notice that is minimised by taking ,
Therefore the minimum possible risk is the Bayes risk, .
Proof of (b):
Proof of (c):
The general case that the Bayes classifier minimises classification error when each element can belong to either of n categories proceeds by towering expectations as follows.
This is minimised by simultaneously minimizing all the terms of the expectation using the classifier
for each observation x.