Model parameters, their prior distributions, and their role in the model.

In most of our analyses, we assume a constant death rate, μ, with an Exponential prior with rate parameter η = 10. In phylodynamic applications, there may be more information to set η, while in macroevolutionary examples one could instead employ an empirical Bayes approach. When there is serial sampling, we adopt an empirical Bayes approach to setting the prior on the sampling rate, ϕ, using a guess at the tree age and the number of tips to obtain . In practice we set ω = 1.17481. In analyses without serial sampling, ϕ = 0. For details on computing , see S1 Text. The sampling fraction at present, Φ0 and the probability of death upon sampling, r are taken to be known a priori. The age of the tree, tor is fixed to the observed height if the tree is data, else it is a variable with the prior determined by the user. For models with n = 100 intervals, we set ζ = 0.0021 for HSMRF-based models and ζ = 0.0094 for GMRF-based models, while for models with other n, we provide code for setting ζ. The GMRF-based model lacks local scale parameters σ. We adopt an empirical Bayes approach to setting the prior on the first log-birth-rate using a guess at the tree age and the number of tips to obtain . In practice we set ξ = 1.17481. In models where the death rate varies, the previously discussed prior on μ serves as the prior on μ1, and the rest of the prior is accomplished via an MRF model exactly as with the birth rate.