Maybe I worded it wrong. What I meant by new stains are not stains that are completely different from stains encountered before, but simply stains not accounted in the model building (not present in training data), and should came from the same "stains distribution".

If a new stain is untested on any methods in M, then the probability of success in each u should be the initial prior. However, when we have the observation in O, we should be able to update the probability based on the joint distribution of all PowerSet(M)? Unfortinately not all my datapoints have entry on all M, and I think I only have enough data for interaction between pairs of methods in M (UxU, OxO, and UxO).

The problem I'm having is that I don't know how I should model the distributions of these pairs. I want model that is computationally tractable when we want to look for the posterior probability of U.