The minimal observable clade size of exchangeable coalescents

Abstract

For Λ-n-coalescents with mutation, we analyse the minimal observable clade size ${\mathit{O}}_{\mathit{n}}$. For a given individual $\mathit{i}\in \{1,\dots ,\mathit{n}\}$, the minimal observable clade size ${\mathit{O}}_{\mathit{n}}(\mathit{i})$ is 1 plus the number of other individuals j such that every non-singleton mutation inherited by i is also inherited by j. We provide asymptotics of ${\mathit{O}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and a recursion for all moments of ${\mathit{O}}_{\mathit{n}}$ for finite n. This statistic gives an upper bound for the minimal clade size (Advances in Applied Probability 37 (2005) 647–662) which is not observable in real data (in contrast to ${\mathit{O}}_{\mathit{n}}$). In applications to genetics, it has been shown to be useful to lower classification errors in genealogical model selection (Freund and Siri-Jégousse, 2020).