I am trying to understand the difference between discriminative models and generative models. One of the helpful answers at stack overflow is here: What is the difference between a generative and a discriminative algorithm?

In the top answer (see the above link), there was a simple example where there are only four data points of the form (x,y). The author of the answer said the following: The distribution p(y|x) is the natural distribution for classifying a given example x into a class y, which is why algorithms that model this directly are called discriminative algorithms. Generative algorithms model p(x,y), which can be transformed into p(y|x) by applying Bayes rule and then used for classification. However, the distribution p(x,y) can also be used for other purposes. For example, you could use p(x,y) to generate likely (x,y) pairs.

I don’t quite understand how one could use p(x,y) to generate likely (x,y) pairs. Could someone help me by giving a specific example of (x,y) pair that is generated by using the joint probability distribution p(x,y)? Also, could someone also tell me why the conditional probability distribution p(y|x) CANNOT be used to generate new pairs? Thank you so much.

I am quite confused about this. Please help me.