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In [[statistical classification]], the '''Bayes error rate''' is the lowest possible error rate for a given class of classifier.<ref name=stat >Fukunaga, Keinosuke (1990) ''Introduction to Statistical Pattern Recognition'' by ISBN 0122698517 pages 3 and 97</ref><ref name=Tumer >K. Tumer, K. (1996) "Estimating the Bayes error rate through classifier combining" in ''Proceedings of the 13th International Conference on Pattern Recognition'', Volume 2, 695–699 </ref>
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A number of approaches to the estimation of the Bayes error rate exist. One method seeks to obtain analytical bounds which are inherently dependent on distribution parameters, and hence difficult to estimate. Another approach focuses on class densities, while yet another method combines and compares various classifiers.<ref name=Tumer />
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The Bayes error rate finds important use in the study  of patterns and [[machine learning]] techniques.
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==Error Determination==
'''MathML'''
In terms of machine learning and pattern classification, the data set can be discretely divided into 2 or more classes. Each element of the dataset is called an '''instance''' and the class it belongs to is called the '''lable'''.
:<math forcemathmode="mathml">E=mc^2</math>
The bayes error rate of the dataset classifier is probability of the classifier to incorrectly classify an instance.
For an n-multiclass classifier, the bayes error rate may be calculated as follows :
<br /><br />
<math>\sum_{i=1}^{i=n}\textstyle \int\limits_{x\in H_{i}}P(C_{i}|x)p(x)\, dx</math>
<br /> <br />
Where x is an instance, C is the class into which an instance is classified, H<math>_{i}</math> is the area/region that the classifier function '''h''' classifies as C<math>_{i}</math>
<br />
A bayes error is non-zero if the distributions of the instances overlap, i.e. a certain instance x can wear more than one lable.


==See also==
<!--'''PNG'''  (currently default in production)
* [[Naive Bayes classifier]]
:<math forcemathmode="png">E=mc^2</math>


==References==
'''source'''
{{Reflist}}
:<math forcemathmode="source">E=mc^2</math> -->


[[Category:Statistical classification]]
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[[Category:Bayesian statistics|Error rate]]


{{Statistics-stub}}
==Demos==
 
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Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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