The Joint Spectrum over Trees under the Kingman coalescent with varying population

The Kingman coalescent process models the genealogy of a sample taken from a large population of individuals who reproduce and die according to models such as the Moran or Wright-Fisher processes. The occurrence and spread of neutral mutations in the sample can be modelled by a Poisson process over sample phylogenies from the Kingman coalescent. We study the joint probability distribution of frequencies for multiple mutations occurring on the same tree. We call this the Joint Spectrum over Trees (JST). We derive a closed-form solution for this joint distribution in the case of two mutations with varying population size. We specialise the result for specific population histories, including constant population size. In the process, we highlight how different averaging procedures can lead to different distributions for the frequency of mutations, even when considering only the frequency of a single mutation. We provide a systematic approximation scheme for the Joint Spectrum over Trees under constant population when the number of samples is large. The exact form of the Joint Spectrum over Trees has implications for genealogical inference with the Kingman coalescent, specifically for the characterisation of tree structure near the root and in parameter inference when the underlying tree structures are unknown. To this end, we also comment on the validity of the independence approximation to the true joint distribution under different population histories.