Kingman Coalescent Tree Moments

Worked Example

Let's trace a small tree with n=4 lineages to manually compute its expected branch length. In a coalescent tree of size 4, there are 3 merger stages, backward in time, when there are k=4, k=3, and k=2 lineages remaining.

1. For k=4, the rate is C(4, 2) = 6. The expected waiting time is 1/6. Total expected branch contribution is 4 × (1/6) = 4/6 = 2/3.
2. For k=3, the rate is C(3, 2) = 3. The expected waiting time is 1/3. Total expected branch contribution is 3 × (1/3) = 1.
3. For k=2, the rate is C(2, 2) = 1. The expected waiting time is 1. Total expected branch contribution is 2 × 1 = 2.
4. Summing these up: 2/3 + 1 + 2 = 11/3 ≈ 3.6666666666666665. This matches the expected total branch length E[L] = 2 H_3 = 2(1 + 1/2 + 1/3) = 11/3.

Experiment

You can interactively explore these properties in the module at https://riano.app/module/kingman-coalescent-tree-moments.

1. Select Sample lineages to 6. Observe that the tree gains more stages and the Expected total branch length increases to approximately 4.566666666666666.
2. Change the Mutation theta to 2. Notice that the Expected segregating sites doubles because mutations fall independently along the branches with rate proportional to theta.
3. Change the Stage probe to 3. This highlights the tree where 3 ancestral lineages remain. The Merger Rate becomes 3.0 and the Mean Wait becomes 0.3333333333333333.

Limitations

The Kingman Coalescent heavily relies on the assumption of a constant panmictic population with neutral mutations (no selection) and no recombination (a single locus). Additionally, if the sample size is only 1 lineage (n=1), there are no mergers, meaning the tree height and branch length are strictly zero. The module appropriately labels stage variables as "Unavailable".

References

For a comprehensive overview of the theoretical coalescent process and its integration into Bayesian phylogenetics, refer to Alexei Drummond's lecture:

• Bayesian Phylogenetics: Lecture on the Coalescent