A–12 APPENDIXprobability distribution, which says how likely we think different values are for the
allele frequency at locus A in the second population. In this example, we’ll use the
likelihood function shown in Figure A.12 for that purpose. The choice of the prior
distribution is made by the investigator, and in some cases it can be based simply
on an intuition about what values are most plausible.
We now sample alleles from the second river. We calculate the likelihood function
for this new sample in the same way as we did earlier for the first river. We then calcu-
late the posterior probability distribution, which is simply the product of the new like-
lihood function and the prior probability distribution. This posterior distribution is the
main goal of a Bayesian analysis. It tells us the relative probability that the actual allele
frequency in the population is any given value. This is the key difference with likeli-
hood estimation (discussed in the previous section), which gives us the likelihood that
we would obtain our sample if the actual allele frequency were a given value.
As with the likelihood function, the posterior probability distribution does not
require us to rely on a single estimate for the allele frequency. It says how probable
any given value is, and we can evaluate that information however we like. If we
want a single estimate of the allele frequency, a good value to use is the one corre-
sponding to the maximum of the posterior distribution. (This is the Bayesian analog
of the maximum likelihood estimate.)
To make the platypus example more specific, imagine that (unknown to us) the
actual frequency in the second population is p 1 = 0.4. After sampling just four alleles,
we find one copy of A 1 and three copies of A 2. Because the sample size is so small, this
does not give us much new information. The likelihood function, again calculated
using Equation A.5, is quite flat (FIGURE A.13A). The posterior probability distribution
is found by multiplying this likelihood function and the prior distribution. Because
the likelihood function is so flat, this posterior distribution is very similar to the prior
distribution. The peak in the posterior distribution corresponds to p 1 = 0.21, which we
can use as the estimate for the frequency of the A 1 allele in the second river.
Now say that we capture more platypuses, and find a total of 8 copies of A 1 and
12 copies of A 2 from the second river. The likelihood function for the data is more
strongly peaked because the larger sample size gives us more confidence in the
actual frequency in the second population (FIGURE A.13B). The posterior distribu-
tion (again given by the product of the prior distribution and the likelihood func-
tion) now estimates that the frequency of A 1 is p 1 = 0.3, closer to the true value of
p 1 = 0.4. Finally, after sampling still more platypuses, we have 37 copies of A 1 and
63 copies of A 2 from the second river. The posterior distribution is now even more
strongly peaked, and nearly centered on the true allele frequency of p 1 = 0.4 (FIGURE
A.13C). Our estimate for the frequency of A 1 is now p 1 = 0.34. With even more data,
the estimate would tend to move even closer to the true value of the allele frequency.
This example shows how the Bayesian approach combines prior information
with new data. Another motivation for using Bayesian methods is simply practi-
cal. Many problems in evolutionary biology, such as estimating phylogenetic trees,
involve extremely complicated likelihood functions. They cannot be analyzed in
the relatively straightforward way that we used in the discussion of allele frequen-
cies in the previous section. A strategy based on Bayesian methods can save the
situation. The basic idea is illustrated in the following example.
Say that we are interested in using DNA sequences to estimate the age of a
node (branching point) in a phylogeny of several species. We use a computer to
randomly sample a possible value for that age from a prior distribution that we
assume. We then calculate the likelihood of our sequence data using that value
(along with assumptions about how rapidly the sequences evolve). We now draw
a second random value for the age of the node, and again calculate the likelihood.23_EVOL4E_APP.indd 12 3/22/17 1:52 PM