A characterisation of the reconstructed birth-death process through time rescaling

TL;DR

This paper introduces a simple time rescaling method for analyzing the reconstructed birth-death process, enabling easier derivation of inter-event times and understanding of population genealogy, including incomplete sampling and scaling limits.

Contribution

It presents a novel, analytic time rescaling approach for the reconstructed process, simplifying the derivation of key distributions and extending results to incomplete sampling and population growth limits.

Findings

Time rescaling yields straightforward inter-event time derivations.

Results apply to incomplete sampling with Bernoulli trials.

Scaling limit as sampling probability approaches zero shows logistic and exponential distributions.

Abstract

The dynamics of a population exhibiting exponential growth can be modelled as a birth-death process, which naturally captures the stochastic variation in population size over time. In this article, we consider a supercritical birth-death process, started at a random time in the past, and conditioned to have n sampled individuals at the present. The genealogy of individuals sampled at the present time is then described by the reversed reconstructed process (RRP), which traces the ancestry of the sample backwards from the present. We show that a simple, analytic, time rescaling of the RRP provides a straightforward way to derive its inter-event times. The same rescaling characterises other distributions underlying this process, obtained elsewhere in the literature via more cumbersome calculations. We also consider the case of incomplete sampling of the population, in which each leaf of the genealogy is retained with an independent Bernoulli trial with probability $\psi$, and we show that corresponding results for Bernoulli-sampled RRPs can be derived using time rescaling, for any values of the underlying parameters. A central result is the derivation of a scaling limit as $\psi$ approaches 0, corresponding to the underlying population growing to infinity, using the time rescaling formalism. We show that in this setting, after a linear time rescaling, the event times are the order statistics of $n$ logistic random variables with mode $\log(1/\psi)$; moreover, we show that the inter-event times are approximately exponentially distributed.