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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?