Quasisymmetric function

In statistical classification the Bayes classifier minimises the probability of misclassification.

Definition

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

$X\mid Y=r\sim P_r$ for $r=1,2,\dots ,K$

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

$\mathcal{ℛ}(C)=\mathrm{P}\{C(X)\ne Y\}.$

The Bayes classifier is

$C^{\text{Bayes}}(x)=\underset{r\in \{1,2,\dots ,K\}}{\mathrm{argmax}}\mathrm{P}(Y=r\mid X=x).$

In practice, as in most of statistics, the difficulties and subtleties are associated with modeling the probability distributions effectively -- in this case, $\mathrm{P}(Y=r\mid X=x)$. The Bayes classifier is one of the useful benchmarks 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.Template:Cn

See also

• Naive Bayes classifier

References

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