[currymj]

if you have an MNIST classifier that just takes in images, and spits out a probability of digits 1-9, that wouldn't necessarily be generative, if it is only capable of modeling P(which digit | all pixels).

But many other types of model would give you a joint distribution P(which digit, all pixels), so would be generative. Even if you only used it for classification.

https://en.wikipedia.org/wiki/Generative_model

I guess these days "generative" must mean "it is used to generate outputs that look like the training data".

But until recently, the meaning had to do with the information in the model, not how it's used.

[zozbot234]

You can derive the latter information (the joint distribution), given the former and a prior over "all pixels"-like data. So, the defining feature of "generative" models is that they feature a prior over their input data?

[Nimitz14]

Generative models model the data, whether that is p(x) or p(x,y) or (x,y,z) etc.

[currymj]

Yes, though maybe not explicitly written down.