Bayesian non-parametric inference for $\Lambda$-coalescents: consistency and a parametric method

TL;DR

This paper develops Bayesian methods for inferring the $ ext{Lambda}$-measure in $ ext{Lambda}$-coalescent models, establishing consistency criteria, analyzing likelihood properties, and proposing algorithms for posterior sampling, with a focus on moment inference.

Contribution

It introduces verifiable criteria for posterior consistency, analyzes likelihood invariance across measures with matching moments, and proposes computational algorithms for Bayesian inference in $ ext{Lambda}$-coalescents.

Findings

Posterior consistency depends on the observation scheme.

Likelihood is constant for measures with matching leading moments.

Pseudo-marginal algorithms are effective for posterior sampling.

Abstract

We investigate Bayesian non-parametric inference of the $\Lambda$-measure of $\Lambda$-coalescent processes with recurrent mutation, parametrised by probability measures on the unit interval. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any non-trivial prior is inconsistent when all observations are contemporaneous. We then show that the likelihood given a data set of size $n \in \mathbb{N}$ is constant across $\Lambda$-measures whose leading $n - 2$ moments agree, and focus on inferring truncated sequences of moments. We provide a large class of functionals which can be extremised using finite computation given a credible region of posterior truncated moment sequences, and a pseudo-marginal Metropolis-Hastings algorithm for sampling the posterior. Finally, we compare the efficiency of the exact and noisy pseudo-marginal algorithms with and without delayed acceptance acceleration using a simulation study.