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| In [[statistical classification]] the '''Bayes classifier''' minimises the [[probability]] of [[misclassification]].<ref name = "PTPR">{{cite book |author = Devroye, L., Gyorfi, L. & Lugosi, G. |year = 1996 |title = A probabilistic theory of pattern recognition| publisher=Springer | isbn=0-3879-4618-7}}</ref>
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| ==Definition==
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| Suppose a pair <math>(X,Y)</math> takes values in <math>\mathbb{R}^d \times \{1,2,\dots,K\}</math>, where <math>Y</math> is the class label of <math>X</math>. This means that the [[conditional distribution]] of ''X'', given that the label ''Y'' takes the value ''r'' is given by
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| :<math>X\mid Y=r \sim P_r</math> for <math>r=1,2,\dots,K</math>
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| where "<math>\sim</math>" means "is distributed as", and where <math>P_r</math> denotes a probability distribution.
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| 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 <math>C: \mathbb{R}^d \to \{1,2,\dots,K\}</math>, 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
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| :<math>\mathcal{R}(C) = \operatorname{P}\{C(X) \neq Y\}.</math>
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| The Bayes classifier is
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| :<math>C^\text{Bayes}(x) = \underset{r \in \{1,2,\dots, K\}}{\operatorname{argmax}} \operatorname{P}(Y=r \mid X=x).</math>
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| In practice, as in most of statistics, the difficulties and subtleties are associated with modeling the probability distributions effectively -- in this case, <math>\operatorname{P}(Y=r \mid X=x)</math>. The Bayes classifier is one of the useful benchmarks in [[statistical classification]].
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| The excess risk of a general classifier <math>C</math> (possibly depending on some training data) is defined as <math>\mathcal{R}(C) - \mathcal{R}(C^\text{Bayes}).</math>
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| Thus this non-negative quantity is important for assessing the performance of different classification techniques. A classifier is said to be [[Consistency (statistics)|consistent]] if the excess risk converges to zero as the size of the training data set tends to infinity.{{cn|date=March 2013}}
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| ==See also==
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| *[[Naive Bayes classifier]]
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| ==References==
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| {{reflist|1}}
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| [[Category:Bayesian statistics]]
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| [[Category:Statistical classification]]
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