Bayesian non-parametric inference for \(\Lambda\)-coalescents: posterior consistency and a parametric method. (English) Zbl 1419.62063

Summary: 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.


62G05 Nonparametric estimation
60G57 Random measures
62F15 Bayesian inference
62G20 Asymptotic properties of nonparametric inference
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
65C05 Monte Carlo methods
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