Model choice--AICc

[AndreMonc]

Although model choice is one step beyond the Momi2 pipeline, I suspect this question will be relevant to many Momi2 users. Does the sample size term "n" in the AICc formula refer to number of sites (SNPs) or number of individuals?

[jackkamm]

I am not too well-versed in the theory behind AIC, so take this with a grain of salt. But, I think that "n" would be the number of SNPs, since the log-likelihood is a sum over SNPs, so it seems natural to consider each SNP as an observation.

HOWEVER, you must remember that the likelihood here is a composite likelihood, not the true likelihood. It makes the approximation that the SNPs are independent. While the MLE will still converge to the correct value (under some assumptions), I am doubtful that the AIC or AICc would be appropriate for a composite likelihood.

[jackkamm]

OK, looking into this more, I think Theorem 1 from this paper shows how to compute AIC for composite likelihood models:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7516723/

In particular, the k in AIC should be replaced by trace(J H^-1) term in Theorem 1. H is the Fisher information and J is the covariance of the score. Both these terms are computed already in momi2/confidence_region.py, e.g. see the godambe() function.