Necessary and sufficient conditions for identifiability in the admixture model
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We consider M SNP data from N individuals who are an admixture of K unknown ancient populations. Let Π_si be the frequency of the reference allele of individual i at SNP s. So the number of reference alleles at SNP s for a diploid individual is binomially distributed with parameters 2 and Π_si. We suppose Π_si=∑_k=1^KF_skQ_ki, where F_sk is the allele frequency of SNP s in population k and Q_ki is the proportion of population k in the ancestry of individual i. I am interested in the identifiability of F and Q, up to a relabelling of the ancient populations. Under what conditions, when Π =F^1Q^1=F^2Q^2 are F^1 and F^2 and Q^1 and Q^2 equal? I show that the anchor condition (Cabreros and Storey, 2019) on one matrix together with an independence condition on the other matrix is sufficient for identifiability. I will argue that the proof of the necessary condition in Cabreros and Storey, 2019 is incorrect, and I will provide a correct proof, which in addition does not require knowledge of the number of ancestral populations. I will also provide abstract necessary and sufficient conditions for identifiability. I will show that one cannot deviate substantially from the anchor condition without losing identifiability. Finally, I show necessary and sufficient conditions for identifiability for the non-admixed case.
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