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Definition

Suppose a pair ( X , Y ) {\displaystyle (X,Y)} takes values in R d × { 1 , 2 , … , K } {\displaystyle \mathbb {R} ^{d}\times \{1,2,\dots ,K\}} , where Y {\displaystyle Y} is the class label of an element whose features are given by X {\displaystyle X} . Assume that the conditional distribution of X, given that the label Y takes the value r is given by ( X ∣ Y = r ) ∼ P r for r = 1 , 2 , … , K {\displaystyle (X\mid Y=r)\sim P_{r}\quad {\text{for}}\quad r=1,2,\dots ,K} where " ∼ {\displaystyle \sim } " means "is distributed as", and where P r {\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 C : R d → { 1 , 2 , … , K } {\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 R ( C ) = P ⁡ { C ( X ) ≠ Y } . {\displaystyle {\mathcal {R}}(C)=\operatorname {P} \{C(X)\neq Y\}.}

The Bayes classifier is C Bayes ( x ) = argmax r ∈ { 1 , 2 , … , K } P ⁡ ( Y = r ∣ X = x ) . {\displaystyle 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, P ⁡ ( Y = r ∣ X = x ) {\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 C {\displaystyle C} (possibly depending on some training data) is defined as R ( C ) − R ( C Bayes ) . {\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 x i {\displaystyle x_{i}} of x {\displaystyle x} to be mutually independent, we get the naive Bayes classifier, where C Bayes ( x ) = argmax r ∈ { 1 , 2 , … , K } P ⁡ ( Y = r ) ∏ i = 1 d P r ( x i ) . {\displaystyle 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 ) {\displaystyle R(h)} , Bayes risk R ∗ {\displaystyle R^{*}} , all possible classes to which the points can be classified Y = { 0 , 1 } {\displaystyle Y=\{0,1\}} . Let the posterior probability of a point belonging to class 1 be η ( x ) = P r ( Y = 1 | X = x ) {\displaystyle \eta (x)=Pr(Y=1|X=x)} . Define the classifier h ∗ {\displaystyle {\mathcal {h}}^{*}} as h ∗ ( x ) = { 1 if  η ( x ) ⩾ 0.5 , 0 otherwise. {\displaystyle {\mathcal {h}}^{*}(x)={\begin{cases}1&{\text{if }}\eta (x)\geqslant 0.5,\\0&{\text{otherwise.}}\end{cases}}}

Then we have the following results:

1. R ( h ∗ ) = R ∗ {\displaystyle R(h^{*})=R^{*}} , i.e. h ∗ {\displaystyle h^{*}} is a Bayes classifier,
2. For any classifier h {\displaystyle h} , the excess risk satisfies R ( h ) − R ∗ = 2 E X [ | η ( x ) − 0.5 | ⋅ I { h ( X ) ≠ h ∗ ( X ) } ] {\displaystyle R(h)-R^{*}=2\mathbb {E} _{X}\left[|\eta (x)-0.5|\cdot \mathbb {I} _{\left\{h(X)\neq h^{*}(X)\right\}}\right]}
3. R ∗ = E X [ min ( η ( X ) , 1 − η ( X ) ) ] {\displaystyle R^{*}=\mathbb {E} _{X}\left[\min(\eta (X),1-\eta (X))\right]}
4. R ∗ = 1 2 − 1 2 E [ | 2 η ( X ) − 1 | ] {\displaystyle R^{*}={\frac {1}{2}}-{\frac {1}{2}}\mathbb {E} [|2\eta (X)-1|]}

Proof of (a): For any classifier h {\displaystyle h} , we have R ( h ) = E X Y [ I { h ( X ) ≠ Y } ] = E X E Y | X [ I { h ( X ) ≠ Y } ] = E X [ η ( X ) I { h ( X ) = 0 } + ( 1 − η ( X ) ) I { h ( X ) = 1 } ] {\displaystyle {\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 ) {\displaystyle R(h)} is minimised by taking ∀ x ∈ X {\displaystyle \forall x\in X} , h ( x ) = { 1 if  η ( x ) ⩾ 1 − η ( x ) , 0 otherwise. {\displaystyle 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 ∗ ) {\displaystyle R^{*}=R(h^{*})} .

Proof of (b): R ( h ) − R ∗ = R ( h ) − R ( h ∗ ) = E X [ η ( X ) I { h ( X ) = 0 } + ( 1 − η ( X ) ) I { h ( X ) = 1 } − η ( X ) I { h ∗ ( X ) = 0 } − ( 1 − η ( X ) ) I { h ∗ ( X ) = 1 } ] = E X [ | 2 η ( X ) − 1 | I { h ( X ) ≠ h ∗ ( X ) } ] = 2 E X [ | η ( X ) − 0.5 | I { h ( X ) ≠ h ∗ ( X ) } ] {\displaystyle {\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): R ( h ∗ ) = E X [ η ( X ) I { h ∗ ( X ) = 0 } + ( 1 − η ( X ) ) I { h ∗ ( X ) = 1 } ] = E X [ min ( η ( X ) , 1 − η ( X ) ) ] {\displaystyle {\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): R ( h ∗ ) = E X [ min ( η ( X ) , 1 − η ( X ) ) ] = 1 2 − E X [ max ( η ( X ) − 1 / 2 , 1 / 2 − η ( X ) ) ] = 1 2 − 1 2 E [ | 2 η ( X ) − 1 | ] {\displaystyle {\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. E Y ( I { y ≠ y ^ } ) = E X E Y | X ( I { y ≠ y ^ } | X = x ) = E [ Pr ( Y = 1 | X = x ) I { y ^ = 2 , 3 , … , n } + Pr ( Y = 2 | X = x ) I { y ^ = 1 , 3 , … , n } + ⋯ + Pr ( Y = n | X = x ) I { y ^ = 1 , 2 , 3 , … , n − 1 } ] {\displaystyle {\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 , arg ⁡ max k P r ( Y = k | X = x ) {\displaystyle h(x)=k,\quad \arg \max _{k}Pr(Y=k|X=x)} for each observation x.

See also

• Naive Bayes classifier

References

1. Devroye, L.; Gyorfi, L. & Lugosi, G. (1996). A probabilistic theory of pattern recognition. Springer. ISBN 0-3879-4618-7. 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. https://dl.acm.org/doi/abs/10.1109/18.243433 ↩