Birth death
This is the birth death model based on Gernhard (2008).
This derivation conditions directly on fixed \(N\) taxa.
The inference is directly on birth diff rate
b-d (strictly positive) and relative death rate
d/b (constrained in [0,1)).
Verified using simulated trees generated by Klaas tree sample. (http://www.klaashartmann.com/treesample/) Sampling proportion not verified via simulation. Proportion set by default to 1, an assignment which makes the expressions identical to the expressions before the change.
The model assumes all tips are sampled at the same time (i.e. no tip dates). If you have tip dates, consider a birth death sampling prior or coalescent tree prior instead.
It has a two parameters, the birth diff rate and relative death rate, which are usually estimated and require a hyper prior (see birth rate prior, death rate prior).
References
Gernhard, T. 2008. The conditioned reconstructed process. Journal of Theoretical Biology Volume 253, Issue 4, 21 August 2008, Pages 769-778 doi:10.1016/j.jtbi.2008.04.005.