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I understand that logistic regression has some nice properties that works well for classification problems, such as the S-curve shape, the output value being between [0,1], and continuous across X. But there are some other functions which also share such properties, for example the Complementary log-log function or even the CDF of Gaussian distribution.

Why is the logistic function most commonly used in classification problems? Is it because the logistic function is relatively easy to implement?

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Why is the logistic function most commonly used in classification problems? Is it because the logistic function is relatively easy to implement?

The logit function is the canonical link function and thus an obvious default (see Wiki on generalized linear models). See this post for some of the properties of canonical link functions which makes them nice.

Do see the post Tim links to.

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    $\begingroup$ The question is confusing. Logistic regression has next to nothing to do with classification. It is a direct probability estimation model. See fharrell.com/post/classification $\endgroup$ – Frank Harrell Mar 5 '18 at 20:46

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