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

We investigate Bayesian non-parametric inference for $\Lambda$-coalescent processes parametrised by probability measures on the unit interval, and provide an implementable, provably consistent MCMC inference algorithm. 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 credibility 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.