You can't marginalize over a parameter in the likelihood without a prior because the units are wrong! The likelihood is the probability of the data given the model, and therefore has units of inverse data. If you want to marginalize out some nuisance parameter, you have to multiply the likelihood by a prior probability distribution for that parameter and then integrate. So, as I like to point out, only Bayesians can marginalize.
Adam Myers and I are using marginalization to get the likelihood for parameters of a distribution for a quantity (in this case exoplanet mass), marginalizing out every individual quantity (mass) estimate. You have to marginalize out the individual mass determinations because they are all terribly biased individually, and it is only the underlying or
uncertainty-deconvolved distribution that you really care about. More soon, especially if we succeed!