The Bayesian model

I'm not going to provide all of the gory details on the Bayesian model. If you're interested you can find most of them in my lecture notes from the Summer Institute in Statistical Genetics last summer.² In fact, I'm only going to describe two pieces of the model.³ First, a little notation:

$$\begin{eqnarray*} n_{11,i} &=& \hbox{\char93 of $A_1A_1$ genotypes} \\ n_{12,... ...box{population index} \\ I &=& \hbox{number of populations} \\ \end{eqnarray*}$$

These are the data we have to work with. The corresponding genotype frequencies are

$$\begin{eqnarray*} x_{11,i} &=& p_{i}^2 + fp_{i}(1-p_{i}) \\ x_{12,i} &=& 2p_{i}(1-p_{i})(1-f) \\ x_{22,i} &=& (1-p_{i})^2 + fp_{i}(1-p_{i}) \end{eqnarray*}$$

So we can express the likelihood of our sample as a product of multinomial probabilities

$$P({\bf n}\vert{\bf p},f) \propto \prod_{i=1}^I x_{11,i}^{n_{11,i}} x_{12,i}^{n_{12,i}} x_{22,i}^{n_{22,i}} \quad .$$

To complete the Bayesian model, all we need are some appropriate priors. Specifically, we so far haven't done anything to describe the variation in allele frequency among populations. Suppose that the distribution of allele frequencies among populations is well-approximated by a Beta distribution. A Beta distribution is convenient for many reasons, and it is quite flexible. Don't worry about what the formula for a Beta distribution looks like. All you need to know is that it has two parameters and that if these parameters are $\pi$ and $\theta$, we can set things up so that

$$\begin{eqnarray*} \mbox{E}(p_{ik}) &=& \pi \\ \mbox{Var}(p_{ik}) &=& \pi(1-\pi)\theta \end{eqnarray*}$$

Thus $\pi$ corresponds to $\bar p$ and $\theta$ corresponds to $F_{st}$.⁴ Figure 1 illustrates the shape of the Beta distribution for different choices of $\pi$ and $\theta$. To complete the Bayesian model we need only to specify priors on $\pi$, $f$, and $\theta$. In the absence of any prior knowledge about the parameters, a uniform prior on [0,1]⁵ is a natural choice.

Figure 1: Shapes of the Beta distribution for different choices of $\pi$ and $\theta$. In the figure captions ``p'' corresponds to $\pi$, and ``theta'' corresponds to $\theta$.