The MultiState option in BayesTraits 3.0 (42); www.evolution.rdg.ac.uk/BayesTraitsV3/BayesTraitsV3.html) was used to identify the model best supported by the data. We initially used the reversible-jump (RJ) procedure, using different sets of priors (tables S3 to S6). Each MCMC simulation was run for 100 million iterations sampled every 1000 generations, with the first 25 million iterations discarded as the burn-in. We assumed convergence when the posterior distribution was approximately normal, and trace of harmonic mean log-likelihoods did not show large jumps across runs. Models visited by the Markov chain were ranked in order of their posterior probability. We also examined rate parameters across the Markov chain plotted in Tracer and the effective sample sizes for the parameters of interest (ESS > 200). Analyses in BayesTraits were also run using polymorphic states (two or more states recorded for a species).

We then compared alternative evolutionary models of social evolution using log BFs. We constructed six different models of social organization. First, we used an ER model, thus simulating equal likelihood for all transitions (Fig. 1A). Second, we fit a six-parameter (three-state scheme: three parameters) SYM model (Fig. 1A), where forward and reverse transitions share the same parameter. Third, rates were allowed to vary freely without constraint to produce a 12-parameter ARD model (Fig. 1A). In the case of a three-state scheme, this model estimated six parameters. The fourth model was an “IC” model where transitions were restricted so that movements were only allowed between solitary and pair living, pair living and unimale groups, and unimale groups and multimale organization (Fig. 1B). In the case of a three-state scheme, transitions were only allowed between solitary and pair living and pair living and group living. The fifth model we tested was the one proposed by Shultz et al. (4), in which transitions were allowed from solitary to multimale and from multimale to pair living and to unimale and back. Transitions from solitary to social are not reversed, such that once a lineage becomes social, it remains so (Fig. 1C).

Last, we tested the model structure with the highest posterior support from the reversible-jump analysis described above. This model is similar to the IC model, but it also allows transitions from multimale to pair living and sets the transition from unimale to pair living equal to zero (Fig. 1D). A schematic representation of each model for the four-state scheme is reported in Fig. 1 (in fig. S1 for the three-state scheme). Each model was evaluated in five independent runs for 50 million iterations sampled every 1000 iterations, with the first 10 million iterations (20%) discarded as the burn-in period. Marginal likelihoods were calculated using stepping stone sampling with 100 samples and 10,000 iterations per sample. The stepping stone sampler estimates the marginal likelihood by placing a number of “stones,” which link the posterior with the prior; the stones are successively heated, forcing the chain from the posterior toward the prior. This procedure provides a more effective estimate of the marginal likelihood. Alternative models were then compared using log BFsLog Bayes factors=2(log marginal likelihood[model 1]  log marginal likelihood [model 2])

The BF shows the weight of evidence to support one model over another (weak evidence, <2; positive evidence, >2; strong evidence, 5 to 10; very strong evidence, >10).

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