It’s certainly my fault for not being clearer.
As far as I know, what you’re suggetsing would be just the likelihood of the data from group s given our estimate of its mean x_s. I wrote this as \sum{\textrm{logp}(x_i | x_s)} above. I’ve seen that used in “leave one group out” likelihoods. I am interested in the best or most canonical way to calculate it.
However, in my view, that probability is insufficient in this case. The parameter associated with the subject (x_s) should also be considered. That is, the likelihood associated with a subject should be the probability of having a subject with the estimated parameter (p(x_s|\mu_0)) times the probability of getting their specific data given that parameter (p(x_i|x_s)).
That’s what I’m trying to calculate.
Is that clearer?