# Bayes classifier

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

## Definition

Suppose a pair $(X,Y)$ takes values in $\mathbb {R} ^{d}\times \{1,2,\dots ,K\}$, where $Y$ is the class label of an element whose features are given by $X$. Assume that the conditional distribution of X, given that the label Y takes the value r is given by $(X\mid Y=r)\sim P_{r}\quad {\text{for}}\quad r=1,2,\dots ,K$ where "$\sim$" means "is distributed as", and where $P_{r}$ 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 $C:\mathbb {R} ^{d}\to \{1,2,\dots ,K\}$, 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 ${\mathcal {R}}(C)=\operatorname {P} \{C(X)\neq Y\}.$
The Bayes classifier is $C^{\text{Bayes}}(x)={\underset {r\in \{1,2,\dots ,K\}}{\operatorname {argmax} }}\operatorname {P} (Y=r\mid X=x).$
In practice, as in most of statistics, the difficulties and subtleties are associated with modeling the probability distributions effectively—in this case, $\operatorname {P} (Y=r\mid X=x)$. The Bayes classifier is a useful benchmark in statistical classification.
The excess risk of a general classifier $C$ (possibly depending on some training data) is defined as ${\mathcal {R}}(C)-{\mathcal {R}}(C^{\text{Bayes}}).$ 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.
Considering the components $x_{i}$ of $x$ to be mutually independent, we get the naive Bayes classifier, where $C^{\text{Bayes}}(x)={\underset {r\in \{1,2,\dots ,K\}}{\operatorname {argmax} }}\operatorname {P} (Y=r)\prod _{i=1}^{d}P_{r}(x_{i}).$

## Properties

Proof that the Bayes classifier is optimal and Bayes error rate is minimal proceeds as follows.
Define the variables: Risk $R(h)$, Bayes risk $R^{*}$, all possible classes to which the points can be classified $Y=\{0,1\}$. Let the posterior probability of a point belonging to class 1 be $\eta (x)=Pr(Y=1|X=x)$. Define the classifier ${\mathcal {h}}^{*}$as ${\mathcal {h}}^{*}(x)={\begin{cases}1&{\text{if }}\eta (x)\geqslant 0.5,\\0&{\text{otherwise.}}\end{cases}}$
Then we have the following results:
* $R(h^{*})=R^{*}$, i.e. $h^{*}$ is a Bayes classifier,
* For any classifier $h$, the excess risk satisfies $R(h)-R^{*}=2\mathbb {E} _{X}\left[|\eta (x)-0.5|\cdot \mathbb {I} _{\left\{h(X)\neq h^{*}(X)\right\}}\right]$
* $R^{*}=\mathbb {E} _{X}\left[\min(\eta (X),1-\eta (X))\right]$
* $R^{*}={\frac {1}{2}}-{\frac {1}{2}}\mathbb {E} [|2\eta (X)-1|]$
Proof of (a): For any classifier $h$, we have ${\begin{aligned}R(h)&=\mathbb {E} _{XY}\left[\mathbb {I} _{\left\{h(X)\neq Y\right\}}\right]\\&=\mathbb {E} _{X}\mathbb {E} _{Y|X}[\mathbb {I} _{\left\{h(X)\neq Y\right\}}]\\&=\mathbb {E} _{X}[\eta (X)\mathbb {I} _{\left\{h(X)=0\right\}}+(1-\eta (X))\mathbb {I} _{\left\{h(X)=1\right\}}]\end{aligned}}$ where the second line was derived through Fubini's theorem
Notice that $R(h)$ is minimised by taking $\forall x\in X$, $h(x)={\begin{cases}1&{\text{if }}\eta (x)\geqslant 1-\eta (x),\\0&{\text{otherwise.}}\end{cases}}$
Therefore the minimum possible risk is the Bayes risk, $R^{*}=R(h^{*})$.
Proof of (b): ${\begin{aligned}R(h)-R^{*}&=R(h)-R(h^{*})\\&=\mathbb {E} _{X}[\eta (X)\mathbb {I} _{\left\{h(X)=0\right\}}+(1-\eta (X))\mathbb {I} _{\left\{h(X)=1\right\}}-\eta (X)\mathbb {I} _{\left\{h^{*}(X)=0\right\}}-(1-\eta (X))\mathbb {I} _{\left\{h^{*}(X)=1\right\}}]\\&=\mathbb {E} _{X}[|2\eta (X)-1|\mathbb {I} _{\left\{h(X)\neq h^{*}(X)\right\}}]\\&=2\mathbb {E} _{X}[|\eta (X)-0.5|\mathbb {I} _{\left\{h(X)\neq h^{*}(X)\right\}}]\end{aligned}}$
Proof of (c): ${\begin{aligned}R(h^{*})&=\mathbb {E} _{X}[\eta (X)\mathbb {I} _{\left\{h^{*}(X)=0\right\}}+(1-\eta (X))\mathbb {I} _{\left\{h*(X)=1\right\}}]\\&=\mathbb {E} _{X}[\min(\eta (X),1-\eta (X))]\end{aligned}}$
Proof of (d): ${\begin{aligned}R(h^{*})&=\mathbb {E} _{X}[\min(\eta (X),1-\eta (X))]\\&={\frac {1}{2}}-\mathbb {E} _{X}[\max(\eta (X)-1/2,1/2-\eta (X))]\\&={\frac {1}{2}}-{\frac {1}{2}}\mathbb {E} [|2\eta (X)-1|]\end{aligned}}$

### 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. ${\begin{aligned}\mathbb {E} _{Y}(\mathbb {I} _{\{y\neq {\hat {y}}\}})&=\mathbb {E} _{X}\mathbb {E} _{Y|X}\left(\mathbb {I} _{\{y\neq {\hat {y}}\}}|X=x\right)\\&=\mathbb {E} \left[\Pr(Y=1|X=x)\mathbb {I} _{\{{\hat {y}}=2,3,\dots ,n\}}+\Pr(Y=2|X=x)\mathbb {I} _{\{{\hat {y}}=1,3,\dots ,n\}}+\dots +\Pr(Y=n|X=x)\mathbb {I} _{\{{\hat {y}}=1,2,3,\dots ,n-1\}}\right]\end{aligned}}$
This is minimised by simultaneously minimizing all the terms of the expectation using the classifier $h(x)=k,\quad \arg \max _{k}Pr(Y=k|X=x)$ for each observation x.