Could we explain the disadvantage of imbalanced data mathematically? Simple setup:

*

*observed response is binary (yes/no, 0/1, positive/negative).

*use logistic regression to model the probability of the response being, say, 1: $P(Y=1|X)$.

*the MLE of the model coefficients enable us to predict the probability by $\frac{1}{1+exp(-X\hat{\beta})}$.

*By checking ROC and AUC, we find the most appropriate threshold of $p_{0}$ such that $\hat{y}=1$ if $\hat{p}>p_{0}$, otherwise $\hat{y}=0$.

In this process, could we mathematically show that $P(Y=1|X)$ is underestimated if we have 10000 0's and 100 1's? How is this bias quantified in formula?

Not all strongly unbalanced data would result in logistic regression underestimating the probability of the minority in a relevant way. If the data is perfectly separated  (and logistic regression is using proper regularization), there will be perfect accuracy.
But, of course, lots of unbalanced data sets will lead to relevantly underestimating the probability of the minority class. And the mathematical proof for each of those data sets would be to simply apply your choice of logistic regression to this data set and check the resulting accuracy.
Here is a link to the paper you mentioned, maybe you can access this one. In there, see Figure 3, they give an intuitive example that shows why we often observe underestimation.
