Exactly! We want to incorporate the association information using Bayes theorem. If you have zero information about the mapping, then your knowledge is invariant under permutations of the data sets (e.g., swapping T0 with T1). That implies that your prior over the associations is uniform over the possible permutations (note that a permutation uniquely specifies an association and vice versa). So, when calculating the correlation, you have to average over all permutations, and the correlation turns out to be identically zero for all possible data. No association means no correlation.

So in the zero information case, we get this weird behavior that isn't what we expect. If the zero information case doesn't work, then we can't expect to get correct answers with only partial information about the associations. We can expect similar strangeness when trying to deal with partial information based on priors about side-effects caused by our hypothetical drug.

If we don't have enough information to construct the model, then our analysis should yield inconclusive results, not weird or backward results. So the problem is to figure out the right way to handle association information.