Evolution, 4th Edition

A STATISTICS PRIMER A–13

We compare the two likelihoods, and then use a probabilistic rule (see https://

en.wikipedia.org/wiki/Metropolis_Hastings_algorithm) that tells us whether to

keep the first value for the node’s age and discard the second, or to do the reverse.

We record the age that is retained, then repeat the process. After thousands (or

even millions) of repetitions, the distribution of ages that we retained will very

closely resemble the posterior probability distribution for the node’s age. We can

use the distribution of retained values to estimate the node’s age (the age that has

the greatest probability) and the confidence interval for that estimate.

This method is one of several that collectively are called Markov Chain Monte

Carlo, often abbreviated as MCMC. In this example, the aim is to estimate a single

quantity (the age of a node). In practice, the method is typically used to do much

more ambitious jobs, such as simultaneously estimating the branching pattern of

the phylogeny, the ages of all its nodes, and the rates of sequence evolution.

Futuyma Kirkpatrick Evolution, 4e

Sinauer Associates

Troutt Visual Services

Evolution4e_A.13.ai Date 01-08-2017 03-01-2017

(A)

(B)

(C)

Prior Likelihood

0.2 0.4 0.6 0.8

Value of p 1

Posterior

Prior Likelihood

Probability density of

p^1

Posterior

Likelihood

Prior

Posterior

FIGURE A.13 Bayesian estimates for the fre-

quency of allele A 1 in a second population of

platypuses. The likelihood function from the

first population (Figure A.12) is used for the

prior distribution. The actual frequency in the

second population is p 1 = 0.4 (the red circle).

(A) A sample of just four alleles from the sec-

ond population has one copy of A 1 and three

copies of A 2. The resulting likelihood function

is quite flat. The posterior distribution (equal to

the product of the prior distribution and the

likelihood function) is very similar to the prior

distribution. The peak in the posterior distribu-

tion, is p 1 = 0.21 (the black diamond). (B) With

a sample of 20 alleles, we have 8 copies of A 1

and 12 copies of A 2. The likelihood function is

more strongly peaked because of the larger

sample size. The posterior distribution now

estimates that the frequency of A 1 is p 1 = 0.3.

(C) With a sample of 100 alleles, we have 37

copies of A 1 and 63 copies of A 2. The poste-

rior distribution is now even more strongly

peaked, and nearly centered on the true allele

frequency of p 1 = 0.4. Our estimate for the

frequency of A 1 is now p 1 = 0.34.

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