Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes

We show that for $0-\alpha$, the Poisson-Dirichlet distribution with parameter $(\alpha, \theta)$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $\alpha =0$

A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process

An important property of Kingman's coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity'. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman's coalescent is the `fastest' to come down from infinity. In this article we study what happens when we counteract this `fastest' coalescent with the action of an extreme form of fragmentation. We augment Kingman's coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $\lambda>0$, into it's constituent elements. We prove that there exists a phase transition at $\lambda=c/2$, between regimes where the resulting `fast' fragmentation-coalescence process is able to come down from infinity or not. In the case that $\lambda<c/2$ we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.Comment: 1 figur

Bayesian nonparametric models of genetic variation

We will develop three new Bayesian nonparametric models for genetic variation. These models are all dynamic-clustering approximations of the ancestral recombination graph (or ARG), a structure that fully describes the genetic history of a population. Due to its complexity, efficient inference for the ARG is not possible. However, different aspects of the ARG can be captured by the approximations discussed in our work. The ARG can be described by a tree valued HMM where the trees vary along the genetic sequence. Many modern models of genetic variation proceed by approximating these trees with (often finite) clusterings. We will consider Bayesian nonparametric priors for the clustering, thereby providing nonparametric generalizations of these models and avoiding problems with model selection and label switching. Further, we will compare the performance of these models on a wide selection of inference problems in genetics such as phasing, imputation, genome wide association and admixture or bottleneck discovery. These experiments should provide a common testing ground on which the different approximations inherent in modern genetic models can be compared. The results of these experiments should shed light on the nature of the approximations and guide future application of these models