I understand that a probabilistic classifier predicts the probability distribution $P(C|X)$. However I do not understand what $P(C|X)$ means.
Is it "$P(C=c|x)$ the probability of belonging to class $c$ given the input $x \in X$"? If so then what is the sample sample?
I have done some researches and the term "calibration" comes up and it says that "$P(C=c|x) = p$ means that the $p$ is the probability of being correct for all inputs with the same output $p$". Is this the definition of the prediction or calibration is just a "metric" we are measuring our prediction?