Time-calibrated phylogenies of living species have been widely used to study the tempo and mode of species diversification. However, it is increasingly clear that inferences about species diversification -extinction rates in particular -can be unreliable in the absence of paleontological data. We introduce a general framework based on the fossilized birth-death process for studying speciation-extinction dynamics on phylogenies of extant and extinct species. The model assumes that phylogenies

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... be modeled as a mixture of distinct evolutionary rate regimes and that a hierarchical Poisson process governs the number of such rate regimes across a tree. We implemented the model in BAMM, a computational framework that uses reversible jump Markov chain Monte Carlo to simulate a posterior distribution of macroevolutionary rate regimes conditional on the branching times and topology of a phylogeny. The implementation we describe can be applied to paleontological phylogenies, neontological phylogenies, and to phylogenies that include both extant and extinct taxa. We evaluate performance of the model on datasets simulated under a range of diversification scenarios. We find that speciation rates are reliably inferred in the absence of paleontological data. However, the inclusion of fossil observations substantially increases the accuracy of extinction rate estimates. We demonstrate that inferences are relatively robust to at least some violations of model assumptions, including heterogeneity in preservation rates and misspecification of the number of occurrences in paleontological datasets.