  A popular line of research in evolutionary biology is the use of
time-calibrated phylogenies for the inference of diversification processes.
This requires computing the likelihood of a given ultrametric tree as the
reconstructed tree produced by a given model of diversification. Etienne &
Rosindell (2012) proposed a lineage-based model of diversification, called
protracted speciation, where species remain incipient during a random duration
before turning good species, and showed that this can explain the slowdown in
lineage accumulation observed in real phylogenies. However, they were unable to
provide a general likelihood formula. Here, we present a likelihood formula for
protracted speciation models, where rates at which species turn good or become
extinct can depend both on their age and on time. Our only restrictive
assumption is that speciation rate does not depend on species status.
  Our likelihood formula utilizes a new technique, based on the contour of the
phylogenetic tree and first developed in Lambert (2010). We consider the
reconstructed trees spanned by all extant species, by all good extant species,
or by all representative species, which are either good extant species or
incipient species representative of some good extinct species. Specifically, we
prove that each of these trees is a coalescent point process, that is, a
planar, ultrametric tree where the coalescence times between two consecutive
tips are independent, identically distributed random variables. We characterize
the common distribution of these coalescence times in some, biologically
meaningful, special cases for which the likelihood reduces to an elegant
analytical formula or becomes numerically tractable.
