You're violating Jaynes's Infinity Commandment:

Never introduce an infinity into a probability problem except as the limit of finite processes!

Hence we need a prior over joint distributions of (X, Y). And yes, I do mean a prior distribution over probability distributions: we are saying that (X, Y) has some unknown joint distribution, which we treat as being drawn at random from a large collection of distributions. This is therefore a non-parametric Bayes approach: the term non-parametric means that the number of the parameters in the model is not finite.