We presented a new Bayesian approach, which provides a powerful tool to estimate rates of speciation and extinction on dated phylogenies based on the likelihood functions of the pure-birth, and birth-death processes while accounting for phylogenetic uncertainty and incomplete taxon sampling. On empirical data, our rate estimation requires a two steps analysis: 1) sampling a posterior distribution of dated phylogenies using a Bayesian molecular clock approach, and 2) estimating posterior diversification rates on these trees. Available programs such as BEAST  and mcmctree  apply in their relaxed molecular clock implementation different birth-death processes as priors on the node ages [53, 54]. When applied to phylogenies obtained under these assumptions, our approach therefore requires that priors on the diversification parameters are specified twice, while ideally divergence times and diversification rates should be estimated jointly. The majority of the diversification processes considered here are however currently not implemented in these programs, and thus need to be estimated independently. Analyses on simulated data show that the default uniform priors on net diversification and extinction fraction in BEAST do not affect the subsequent rate estimates (relative rate variations lower than 3%; Additional file 3). Alternatively, researchers could choose to run molecular clock analyses in which the prior on the node ages is not based on a birth-death process, but modeled using uniform or Dirichlet distributions [55, 56], as implemented in e.g. Multidivtime  and PhyloBayes .
The range of models implemented can be used to detect a number of different scenarios of rate variation, including specific events of rate increase or decrease, continuous rate variation through time, and clade-specific diversification rates, while simultaneously accounting for taxon sampling. In addition, the Bayesian framework extends the use of the birth-death models beyond the simple rate estimation allowing the comparison of alternative scenarios of diversification for hypothesis testing.
The Bayes factor test computed via thermodynamic integration has shown to represent a reliable and powerful approach to choose among different models of diversification. BF can be applied to compare non-nested models and does not require to be explicitly corrected for the number of model parameters. For these properties it is particularly suitable not only to select the best model in a Bayesian framework, but also to compare specific hypotheses. The power of Bayes factors to detect the correct number of rates predominantly depends on the magnitude of the rate shifts. Similarly, different birth-death and pure-birth processes can generate diversification patterns that might be difficult to distinguish [4, 8]. For instance, an increase in the net diversification rate can be the result of an increased speciation rate in the absence of extinction or a high extinction rate in a constant rate birth-death process. Nevertheless, we found that the Bayes factors test has the power to discern between most of such scenarios.
Analyses on simulated data show that for both speciation and extinction rates the posterior estimates are accurate. However, the width of the 95% HPD intervals also highlights the sometimes considerable uncertainty in the parameter estimates, especially in case of small phylogenies. Our simulations have shown that this uncertainty is most pronounced with a low relative extinction rate, in which case extinction tends to be overestimated [see also , and estimates have a wide 95% credibility interval. This corroborates previous studies that pointed out that the estimation of extinction rates from molecular phylogenies with reasonable degrees of confidence is very problematic [7, 12, 59, 60]. It should be noted, however, that the wide credibility intervals reflect not only the uncertainties of the parameter estimation, but also the uncertainty of the data (i.e. the node ages). Estimates of the speciation rate on the other hand appear to be more robust. In contrast to Paradis , we find that the posterior estimate of λ has a small relative error, even under high relative extinction rates, suggesting that the accuracy in the estimates of λ might be decoupled from the relative rate of extinction. The pure-birth and birth-death models with taxon sampling are found to provide accurate rate estimates, although a poor sampling yields substantially wider 95% HPDs. Finally, the pure-birth process with rate-shifts tends to slightly underestimate the true variation of λ, as a consequence of accounting for the uncertainty of the time of rate shift.
The approach provides a very flexible framework for customized analyses and hypothesis testing such as predefined times of rate shift or constrained parameter values. We have shown with the diversification of Chondrostoma, that the pure-birth model with variable rates can be easily adapted to test for specific hypotheses, running the analysis on fixed time frames defined for example on the basis of geological events or climate changes. The implementation of clade-specific rate estimation further extends the range of options for hypothesis testing, and its application on the radiation of Lupinus showed that it can be used to identify differential rates of diversification between clades. In particular, the option to account for clade-specific sampling biases provides an important feature, as complete taxon sampling is often difficult to achieve, especially for species-rich groups.
Finally, with the F-model, we introduce a new approach to test hypotheses in a meta-analysis framework, and extend the focus from the taxon-specific rate of diversification to a parameter that might be linked to the difference between e.g. geologic periods or geographic regions. The current implementation allows to compare hypotheses with two rates, assigned to either fixed time frames or clades, while accounting for clade-specific taxon sampling. Because the F parameter is constant across data sets, we assume that the magnitude of the rate variation in time or between clades is equal among all data sets. While this certainly represents a simplification of the diversification process, the F-model allows the analyses of potentially many data sets, limiting the number of parameters, and yielding an estimation of general trends across different taxonomic groups. Moreover, even relatively small rate variations can be detected if supported by a sufficient number of data sets.