ABSTRACT

Equation 4 describes how to find the posterior distribution of the rate parameters on a single tree. To account for the influence of phylogenetic uncertainty on Q's posterior distribution, it is necessary to integrate Equation 4 over all possible trees as well as over rate coefficients. Write:

to denote the posterior probability of Q integrated over trees, where it is understood that the integral for trees is over all possible branch lengths and parameters of the model of sequence evolution (Equation 2). Equation 5 says that the probability of a given set of rate parameters in Q is their proportion of the total likelihood, now calculated over all possible trees and sets of rates. By sampling repeatedly from a Markov chain that moves among phylogenetic trees and values of Q, an estimate of the joint distribution of the rate parameters across trees can be obtained. This joint density distribution then accommodates phylogenetic uncertainty in the sense that all of the variability (uncertainty) attributable to trees is included in the posterior density.