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Apologies in advance, this question may be completely obvious.

Say, I didn't have a family tree for two individuals. Could I calculate their coefficient of inbreeding? If not, is there a good proxy for it or an alternative way to calculate it?

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Yes.

These days inbreeding coefficients are generally computed from observed gene frequencies, not from pedigrees.

From this review:

Inbreeding coefficients are defined in terms of the probability of identity in state of different pairs of genes. Here the probability of identity in state is simply the probability that two genes are of identical allelic type.

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