Bayes classifier

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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

Suppose a pair ${\displaystyle (X,Y)}$ takes values in ${\displaystyle \mathbb {R} ^{d}\times \{1,2,\dots ,K\}}$, where ${\displaystyle Y}$ is the class label of an element whose features are given by ${\displaystyle X}$. Assume that the conditional distribution of X, given that the label Y takes the value r is given by

where "${\displaystyle \sim }$" means "is distributed as", and where ${\displaystyle P_{r}}$ 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 ${\displaystyle C:\mathbb {R} ^{d}\to \{1,2,\dots ,K\}}$, 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, ${\displaystyle \operatorname {P} (Y=r\mid X=x)}$. The Bayes classifier is a useful benchmark in statistical classification.

The excess risk of a general classifier ${\displaystyle C}$ (possibly depending on some training data) is defined as ${\displaystyle {\mathcal {R}}(C)-{\mathcal {R}}(C^{\text{Bayes}}).}$ 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 ${\displaystyle x_{i}}$ of ${\displaystyle x}$ to be mutually independent, we get the naive Bayes classifier, where

Properties

Proof that the Bayes classifier is optimal and Bayes error rate is minimal proceeds as follows.

Define the variables: Risk ${\displaystyle R(h)}$, Bayes risk ${\displaystyle R^{*}}$, all possible classes to which the points can be classified ${\displaystyle Y=\{0,1\}}$. Let the posterior probability of a point belonging to class 1 be ${\displaystyle \eta (x)=Pr(Y=1|X=x)}$. Define the classifier ${\displaystyle {\mathcal {h}}^{*}}$as

Then we have the following results:

Proof of (a): For any classifier ${\displaystyle h}$, we have

where the second line was derived through Fubini's theorem

Notice that ${\displaystyle R(h)}$ is minimised by taking ${\displaystyle \forall x\in X}$,

Therefore the minimum possible risk is the Bayes risk, ${\displaystyle R^{*}=R(h^{*})}$.

Proof of (b):

Proof of (c):

Proof of (d):

General case

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 ${\displaystyle h(x)=k,\quad \arg \max _{k}Pr(Y=k|X=x)}$ for each observation x.

See also

• Naive Bayes classifier

References