Coalescent Point Processes are random ultrametric trees yielding interesting results for evolutionary biology.

More specifically, a Coalescent Point Process with stem age T and scale function F is a sequence of iid. random variables (H_i)_{i\geq 1} \sim H, such that F(t) = \frac{1}{\mathbb P(H>t)} killed at the first value H_i>T.

Getting a CPP from a splitting tree

To me the most magic thing about CPP is that you can actually find a distribution for H so that the CPP accurately represent the descent of an individual following a biologically relevant population dynamics model. (Cf proposition 5 in Lambert & Stadler, 2013)

As an example, consider the linear birth-death process with constant birth rate b and constant death rate d.

References

• Lambert, Amaury. “Probabilistic Models for the (Sub)Tree(s) of Life.” Brazilian Journal of Probability and Statistics 31, no. 3 (August 2017): 415–75. https://doi.org/10.1214/16-BJPS320.
• Lambert, Amaury. “The Allelic Partition for Coalescent Point Processes.” ArXiv:0804.2572 [Math], April 16, 2008. http://arxiv.org/abs/0804.2572.
• Lambert, Amaury, and Tanja Stadler. “Birth–Death Models and Coalescent Point Processes: The Shape and Probability of Reconstructed Phylogenies.” Theoretical Population Biology 90 (December 2013): 113–28. https://doi.org/10.1016/j.tpb.2013.10.002.

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