8.4: Using Bayesian MCMC to estimate parameters of the Mk model
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We can also analyze this model using a Bayesian MCMC framework. We can modify the standard approach to Bayesian MCMC (see chapter 2):

Sample a starting parameter value, q, from its prior distributions. For this example, we can set our prior distribution as uniform between 0 and 1. (Note that one could also treat probabilities of states at the root as a parameter to be estimated from the data; in this case we will assign equal probabilities to each state).

Given the current parameter value, select new proposed parameter values using the proposal density Q(q′q). For example, we might use a uniform proposal density with width 0.2, so that Q(q′q) U(q − 0.1, q + 0.1).

Calculate three ratios:

a. The prior odds ratio, R_{prior}. In this case, since our prior is uniform, R_{prior} = 1.

b. The proposal density ratio, R_{proposal}. In this case our proposal density is symmetrical, so R_{proposal} = 1.

c. The likelihood ratio, R_{likelihood}. We can calculate the likelihoods using Felsenstein’s pruning algorithm (Box 8.1); then calculate this value based on equation 2.26.


Find R_{accept} as the product of the prior odds, proposal density ratio, and the likelihood ratio. In this case, both the prior odds and proposal density ratios are 1, so R_{accept} = R_{likelihood}

Draw a random number u from a uniform distribution between 0 and 1. If u < R_{accept}, accept the proposed value of both parameters; otherwise reject, and retain the current value of the two parameters.

Repeat steps 25 a large number of times.
We can run this analysis on our squamate data, obtaining a posterior with a mean estimate of q = 0.001980785 and a 95% credible interval of 0.001174813 − 0.003012715.