We can get the prediction probabilities of a binary classifier from sklearn's API using the predict_proba method. Is it reasonable to expect that the shape of a histogram plotted for the prediction probabilities of let's say the '1' class to approximate a normal distribution? What is the statistical theory that allows for this? I noticed this instance for a logistic regression model I trained.
The landscape of the output probabilities depends entirely on the training data. If the data itself is sampled from a normal distribution, then the learned probabilities will reflect that. Otherwise, no.
Therefore, in the general case, no, it is not reasonable to expect the shape of a histogram plotted from the prediction probabilities to approximate a normal distribution.