In statistical classification, the Bayes classifier is the classifier having the smallest probability of misclassification of all classifiers using the same set of features.[1]

Definition

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Suppose a pair   takes values in  , where   is the class label of an element whose features are given by  . 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.[2]

Considering the components   of   to be mutually independent, we get the naive Bayes classifier, where  

Properties

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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:

  1.  , i.e.   is a Bayes classifier,
  2. For any classifier  , the excess risk satisfies  
  3.  
  4.  

Proof of (a): For any classifier  , we have   where the second line was derived through Fubini's theorem

Notice that   is minimised by taking  ,  

Therefore the minimum possible risk is the Bayes risk,  .

Proof of (b):  


Proof of (c):  

Proof of (d):  

General case

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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.

See also

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References

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  1. ^ Devroye, L.; Gyorfi, L. & Lugosi, G. (1996). A probabilistic theory of pattern recognition. Springer. ISBN 0-3879-4618-7.
  2. ^ Farago, A.; Lugosi, G. (1993). "Strong universal consistency of neural network classifiers". IEEE Transactions on Information Theory. 39 (4): 1146–1151. doi:10.1109/18.243433.