Bayesian molecular dating as a "doubly intractable" problem

1This study focuses on a conceptual issue with Bayesian inference of divergence times using Markov chain Monte Carlo. The influence of fossil data on the probabilistic distribution of trees is the crux of the matter considered here. More specifically, among all the phylogenies that a tree model (e.g., the birth-death process) generates, only a fraction of them \"agree\" with the fossil data at hands. Bayesian inference of divergence times using Markov Chain Monte Carlo requires taking this fraction into account. Yet, doing so is challenging and most Bayesian samplers have simply overlooked this hurdle so far, thereby providing approximate estimates of divergence times and tree process parameters. A generic solution to this issue is presented here. This solution relies on an original technique, the so-called exchange algorithm, dedicated to drawing samples from \"doubly intractable\" distributions. A small example illustrates the problem of interest and the impact of the approximation aforementioned on tree parameter estimates. The analysis of land plant sequences and multiple fossils further illustrates the importance of proper mathematical handling of calibration data in order to derive accurate estimates of node age.