Generative algorithms for learning classifiers use training data to separately estimate a probability model for each class. New items are classified by comparing their probabilities under these models. In contrast, discriminative learning algorithms try to find classifiers that perform well on all the training data.
We show that there is a learning problem that can be solved by a discriminative learning algorithm, but not by any generative learning algorithm. This statement is formalized using a framework inspired by previous work of Goldberg.