For extreme error cases, however, mean tree length starts out much too high in the inferred trees, becomes correct around the point where declared error is equal to true error, and then becomes much too low with further correction.
Branch length priors of exponential mean 1, 0.1 and 0.01 resulted in unrealistically long tree lengths; results from the smallest (exponential mean of 0.001) branch-length prior analysis are presented (mean tree length = 0.28, LCL = 0.26, UCL = 0.30), although there were no significant differences in tree topologies.
The tree used to simulate the data was a strictly bifurcating and balanced tree topology (similar to that in Fig. 2) with branch lengths chosen at random for each simulation from a uniform distribution where the mean tree length, total of all branch lengths, was set a priori to 1 or 3.
We then tried a lower prior rate of evolution, with the mean prior tree length E t) set to 1.5 by setting α = 3 and β = 2 (the gamma distribution was discretised into 50 categories).
For the overall rate of evolution for both characters, we first tried a high rate of evolution, assuming that characters could evolve rapidly: the mean prior tree length E(T) was set to 10 and the SD to 1, by setting α = 100 and β = 10.
However, there, it has a much longer length than in the posterior mean tree (by ∼0.011) because this length is obtained by averaging only the lengths of this edge when it occurs in the sample, neglecting the contributions of the alternative edges.
Results were evaluated after a burn-in period of 10% (150,000 generations) and convergence was achieved (PSRF= 1.00) for all model parameters estimated, including tree length (mean = 18.8), α = 2.28 and the proportion of invariant sites (4%), the amino acid model (Blosum), and the tree topology (see results).
Chains were stopped when the maximum discrepancy in bipartition frequencies and several additional summary variables (including the alpha parameter for across-site rate variation, tree length, mean posterior log-likelihood) between the two chains dropped below 0.1, and the effective sizes of the summary variables were all more than 100, as recommended by the authors.
Convergence was achieved (Potential Scale Reduction Factor, PSRF=1.00) for all model parameters estimated, including tree length (mean-1.79), the amino acid model (Wag, with posterior probability=1.00), and the tree topology.
Clearly, the shorter edge length of the posterior mean tree better accounts for the uncertainty, and the consensus tree appears, in contrast, to have overestimated branch lengths.
Substitution rates at each site were divided by the estimated tree length across the entire gene to control for the mean substitution rate of the encoding gene.
