... snails.¹

It may well be characteristic of many hermaphroditic animal parasites. You should also know that I just lied. I do that a lot, so you should be on the watch for it. In this case I lied because the form of self-fertilization I'm going to describe actually isn't the most extreme form of selfing possible. That honor belongs to gametophytic self-fertilization in homosporous plants. The offspring of gametophytic self-fertilization are uniformly homozygous at every locus in the genome. For more information, if you're interested, see [1]

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... changing.²

This is analogous to stopping the calculation and re-calculation of allele frequencies in the EM algorithm when the allele frequency estimates stop changing.

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... course.³

Unfortunately, I'll also be using hats to denote estimates of unknown parameters, as I did when discussing maximum-likelihood estimates of allele frequencies. I apologize for using the same notation to mean different things, but I'm afraid you'll have to get used to figuring out the meaning from the context. Believe me. Things are about to get a lot worse. Wait until I tell you how many different ways population geneticists use a parameter $f$ that is commonly called the inbreeding coefficient.

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... observe⁴

Important note: I'm assuming that we know the actual genotype frequencies in the population here. In practice, we don't know them. We have to estimate them from the sample, so the frequency of heterozygotes in our sample isn't necessarily the same as the frequency of heterozygotes in our populations. Calling $\hat x_{12}$ is, therefore, a little misleading, but that's what we'll do for the time being.

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... expectations.⁵

$f$ can be negative if there are more heterozygotes than expected, as might be the case if cross-homozygote matings are more frequent than expected at random.

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... defined.⁶

See paragraphs above describing the population and equilibrium inbreeding coefficient.

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... history.⁷

Notice that we could have had each allele mutate independently to $A_2$.

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... descent.⁸

Systematists in the audience will recognize this as the problem of homoplasy.

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... descent.⁹

Notice that if we adopt this definition for $f$ it can only take on values between 0 and 1. When used in the sense of a population or equilibrium inbreeding coefficient, however, $f$ can be negative.

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