You would specify a prior for *parameters*. You haven't mentioned any! 

You need to write some model for how the *distribution* of $Y$ relates to $X$, so your first step is to identify a distribution for $Y$ (it's 0-1, so there's an obvious first choice there) and then write a model for its parameters in terms of $X$.

Once you have that model, you need to think about priors on the parameters. Again, a first attempt should be relatively simple, then make it more complicated if you think it needs to be.