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I know the mean square error formula and how to compute it. When we talk about a regression we can compute the mean square error. However can we talk about a MSE for a classification problem and how to compute it?

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Many classifiers can predict continuous scores. Often, continuous scores are intermediate results that are only converted to class labels (usually by threshold) as the very last step of the classification. In other cases, e.g. posterior probabilities for the class membership can be calculated (e.g. discriminant analysis, logistic regression). You can calculate the MSE using these continuous scores rather than the class labels. The advantage of that is that you avoid the loss of information due to the dichotomization.
MSE is often called Brier's score in classification context.

In this paper we explain it as part of a more general framework: C. Beleites, R. Salzer and V. Sergo:
Validation of Soft Classification Models using Partial Class Memberships: An Extended Concept of Sensitivity & Co. applied to Grading of Astrocytoma Tissues
Chemom. Intell. Lab. Syst., 122 (2013), 12 - 22.

How to compute it: if you work in R, th implementation is package "softclassval", http:/softclassval.r-forge.r-project.org.

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I don't quite see how... successful classification is a binary variable (correct or not), so it is difficult to see what you would square.

Generally classifications are measured on indicators such as percentage correct, when a classification that has been estimated from a training set, is applied to a testing set that was set aside earlier.

Mean square error can certainly be (and is) calculated for forecasts or predicted values of continuous variables, but I think not for classifications.

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