The probability distribution of the reconstructed phylogenetic tree with occurrence data

We consider a homogeneous birth-death process with incomplete sampling. Three successive sampling schemes are considered. First, individuals can be sampled through time and included in the tree. Second, they can be occurrences which are sampled through time and not included in the tree. Third, individuals reaching present day can be sampled and included in the tree. Upon sampling, individuals are removed (i.e. die).\n\nThe outcome of the process is thus composed of the reconstructed evolutionary tree spanning all individuals sampled and included in the tree, and a timeline of occurrence events which are not placed along the tree. We derive a formula allowing one to compute the joint probability density of these, which can readily be used to perform maximum likelihood or Bayesian estimation of the parameters of the model.\n\nIn the context of epidemiology, our probability density allows us to estimate transmission rates through a joint analysis of epidemiological case count data and phylogenetic trees reconstructed from pathogen sequences. Within macroevolution, our equations are the basis for taking into account fossil occurrences from paleontological databases together with extant species phylogenies for estimating speciation and extinction rates. Thus, we provide the theoretical framework for bridging not only the gap between phylogenetics and epidemiology, but also the gap between phylogenetics and paleontology.