Hi Ou,
I actually pondered about this exact question during the time of writing. So I think it’s a great question, that questions details that we take for granted. I believe this depends on the “problem” you’re trying to solve. There are two types of ML problems; generative and discriminative. The problem I’m trying to solve in this task is a discriminative problem.

Generative problems try to find the true P(X,Y) (which is difficult) where discriminative problems try to find P(Y|X) (which is simpler). This way, we can substitute D=X,Y for generative problems, but for discriminative problems, it’ll be D=Y|X.

So as you can see, for generative problems, yes, the likelihood will be P(X,Y|theta), where for the discriminative problems, it is P(Y|X,theta). Hope this helps!