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

Abstract

We investigate Bayesian non-parametric inference of the Λ-measure of Λ-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 ∈ N is constant across Λ-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. 1 arXiv:1512.00982v5 [stat.ME] 23 Jan 2017 and incorrect inference. Consequently, likelihood-based inference for Λ-coalescents has also been an active area of research [Birkner and Blath, 2008, Birkner et al., 2011, Koskela et al., 2015. A review of Λ-coalescents and their use in population genetic inference can be found in [Birkner and Blath, 2009], and Steinrücken et al. [2013] present a review of Beta-coalescent models for marine species. In this paper we make the following contributions:1. We provide the first non-parametric analysis of inferring Λ-measures from genetic data.Our method is also the first instance of inferring Λ using the Bayesian paradigm.2. We prove inconsistency of the posterior in full generality when data is contemporaneous, and give verifiable criteria for consistency when the data set forms a time series.3. We present an implementable parametrisation of the non-parametric inference problem by quotienting the infinite dimensional space M 1 ([0, 1]) in a suitable, data-driven way. We believe this quotienting approach to have utility in infinite dimensional inference beyond the context of this work.4. We implement a pseudo-marginal MCMC algorithm for sampling the posterior, and provide an illustrative simulation study which demonstrates the feasibility of the algorithm for inference.

The multifurcating skyline plot

“…In the general case the relevant parameter space (i.e. the space of probability measures on [0,1]) is too large to enable direct inference of Λ (see Koskela, Jenkins, and Spanò 2018) for more advanced techniques). Therefore, we will limit ourselves here to the one-parameter family of italicBeta(2-α,α)-coalescents, with α∈(0,2).…”