[hervature]

This is the problem when people use technical terms loosely and interchangeably with their English definitions. Generative model classifiers are precisely as you describe. They model a joint distribution that one can sample.

GANs cannot even fit this definition because it is not a classifier. It is composed of a generator and a discriminator. The discriminator is a discriminative classifier. The generator is, well, a generator. It has nothing to do with generative model classifiers. Then you get some variation of neural network generator > model that generates > generative model. This leads to confusion.