PRIOR PROBABILITIES OF STRATIGRAPHIC GAPS IMPLIED BY PHYLOGENIES

Likelihood tests of alternative phylogenetic scenarios allow us to assess hypotheses given the joint probabilities of observed morphologies and stratigraphic ranges given some tree and a range of evolutionary and preservation models. However, some conceptual difficulties arise when two trees have different numbers of hypothesized observed ancestors and/or polytomies, and thus have different numbers of freely varying branches. Reducing branches reduces the likelihoods of trees given morphology by reducing possible evolutionary pathways. Conversely, reducing branches (particularly by hypothesizing observed ancestors) increases the likelihoods of trees given stratigraphy by reducing possible sampling gaps. An alternative is to estimate prior probabilities on numbers of gaps in a tree. Although the single most probable gap for any one lineage is no gaps, the expected number of gaps for N lineages with preservation rate R approximates a gamma distribution with shape=N and scale=1/R. For any one lineage, this includes unsampled ancestors, which obviates a need to accommodate speciation rates. A small correction is needed to take into account expected “shared” gaps due to shared unsampled common ancestors. Because we can empirically estimate sampling intensities and opportunities, one can take into account variable sampling over time and among taxa. In the former case, we can even allow for some completely unsampled regions or environments; in the latter case, we can use among-lineage variation in sampling rates (typically following a lognormal distribution) to model a hyperprior distribution for R. Finally, because we can estimate probabilities of lineages being in different geographic regions or environments, the prior probabilities can vary on different trees that allow for easier (or more difficult) scenarios for "hiding" lineages. In addition to obviating the issue of different model complexities for the same hypothesis, another intuitive benefit of this approach is that some number of gaps will almost always be more probable than no stratigraphic gaps, even though the most likely scenario for any one lineage (and thus the whole clade) is no gaps. This in turn can increase the posterior probability of deeper divergences that otherwise are less likely.