bias variance decomposition for classification problem It is given that:

MSE = bias$^2$ + variance

I can see the mathematical relationship between MSE, bias, and variance. However, how do we understand the mathematical intuition of bias and variance for classification problems (we can't have MSE for classification tasks)? 

I would like some help with the intuition and in understanding the mathematical basis for bias and variance for classification problems. 

Any formula or derivation would be helpful.
My opinion is that the bias variance trade off is rooted in the Uncertainty principle. It behaves absolutely the same.