Conditioning on subsets of the data: applications to ascertainment and other genetic problems

I here consider the question of when to formulate a likelihood over the whole data set, as opposed to conditioning the likelihood on subsets of the data (i.e., joint vs. conditional likelihoods). I show that when certain conditions are met, these two likelihoods are guaranteed to be equivalent, and thus that it is generally preferable to condition on subsets, since that likelihood is mathematically and computationally simpler. However, I show that when these conditions are not met, conditioning on subsets of the data is equivalent to introducing additional df into our genetic model, df that we may not have been aware of. I discuss the implications of these facts for ascertainment corrections and other genetic problems.