# The existence and abundance of ghost ancestors in biparental populations

The existence and abundance of ghost ancestors in biparental populations

Simon Gravel, Mike Steel
(Submitted on 15 Jan 2014)

In a randomly-mating biparental population of size N there are, with high probability, individuals who are genealogical ancestors of every extant individual within approximately log2(N) generations into the past. We use this result of Chang to prove a curious corollary under standard models of recombination: there exist, with high probability, individuals within a constant multiple of log2(N) generations into the past who are simultaneously (i) genealogical ancestors of {\em each} of the individuals at the present, and (ii) genetic ancestors to {\em none} of the individuals at the present. Such ancestral individuals – ancestors of everyone today that left no genetic trace — represent ghost’ ancestors in a strong sense. In this short note, we use simple analytical argument and simulations to estimate how many such individuals exist in Wright-Fisher populations.

## 7 thoughts on “The existence and abundance of ghost ancestors in biparental populations”

1. Pingback: A week of links - Evolving Economics

2. It’s been a while since I worked through Baird et al, but perhaps the difference is the conditioning on the individual being a genealogical ancestor to the entire population, rather than just asking about the survival of their genome (unconditional on this).

3. Thanks for this comment. Indeed, this is very related work that we had missed, and we will modify our manuscript to cite this source. Our simulations are more realistic than Baird et al: we simulate diploid genomes in a finite population, where they use an approximate model that is accurate in the infinite population size limit. The Baird et al approximation is quite reasonable, however, and we show convergence to their predicted functional form for large enough populations. In practice, their asymptotic results break down only for quite small populations or very long times, so the predicted Baird et al. qualitative form remains valid in most practical cases. We find about half as many ghosts’ as they do, though, so there may be some more important differences in the quantitative predictions of both models. I have to make sure that it’s not a mundane difference before making a claim here, though.
Thanks again for the heads up!

• No problem happy to help. I’ll give both a read/reread in the nearly future, as I’m keen to understand this stuff.