Great discussion in Monte Carlo Seminar with Rannala on the two rate estimation papers. See in-text notes, but focus on mechanisms of speciation and extinction as intrinsic (lineage specific) or extrinsic (perhaps spatially specific or global).
Calculation of the likelihood function for the rates seems straight forward. Start by imaging the pure-birth model, simply allow branch lengths as draws from the exponential distribution of parameter lambda. Allowing extinction: probability a lineage speciates (as before) and then survives to present day. In a density dependent model perhaps this second probability can be easily modified by the expected number of species (really needs to be the distribution, but that’s straight forward). Bruce suggests this calculation hasn’t been done. It certainly doesn’t seem to appear in use in the literature. Believe the structure of the data needed is the same as the birth-death. For pure-birth, the data is just the branch lengths. Now the data is the branch length plus the time until present:
Looking at a given branch, the probability that it has descendants in the pure birth-death process is given by the classic branching process results, i.e. Galton-Watson process from Feller (wish I had epub versions of all my textbooks right now…) Note we want the time-dependent solution, not just the escape probability at $ T = $,
\[ P(t, T) = \frac{\lambda-\mu}{\lambda - \mu e^{-(\lambda-\mu)(T-t)}} \]
The probability of zero lineages at time $ t$ is then the complement of this. The probability of $ i > 0 $ lineages at time $ t$ is:
[latex] P(0,t) (1-u_t)u_t^{i-1} \
u = [/latex]
Why is this true? Going through Kendall 1948 papers which just reference a footnote:
*(Added in proof.) Dr Bartlett has pointed out to me that the result (27) is stated by N. Arley in Acta Math (1945) 76 298-9
hmm… calculations for another day.