  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.
