On Some Approximate Inference Approaches in Population Genetics

The coalescent is the primary probabilistic model for modeling genealogies. However, in practice, the exact inference is typically impossible. Several approximated inference approaches have been proposed. Despite their popularity, some aspects of their mechanism remain uncovered. This thesis is devoted to some of these questions. In what follows, we first obtain new theoretical results about the sequentially Markov coalescent (SMC). In an inferential setting, we examine the well-known pairwise sequentially Markov coalescent(PSMC) algorithm for inferring local ancestry using genetic variation data. We prove that the posterior distribution of (a slight variant of) PSMC contracts, up to log factors, at a nearly parametric rate to the underlying true parameter. Second, we study the ergodicity properties of the underlying SMC process itself. We prove via a coupling argument that the underlying discrete jump chain of the process is geometrically ergodic. Establishing similar results for the continuous-time process is harder due to the dependence between the level and jump rate, and, in particular, an absorbing state at zero. We outline some ways forward by drawing parallels between SMC and a model of internet traffic which turns out to have close connections. Next, we investigate the Li-Stephens(LS) model. Interpreting the LS model as a changepoint detection approach, we treat the recombination rate and the mutation rate as two tuning parameters. By appealing to a result in the changepoint detection literature and several results from convex analysis, we derive a new, efficient algorithm for finding the complete solution surface of both haploid and diploid variants of the LS algorithm. The simulation studies demonstrate our algorithm can be viewed as characterizing the trade-off between recombination and mutation. Lastly, we provide a shrinkage estimator for estimating population size history. The estimator based on sample frequency spectrum and PSMC are two popular estimators for estimating demography, however, these two estimators both have their own deficiencies. We consider the linear combination of these two estimators and derive the optimal coefficient for constant component-wise shrinkage coefficient and variable component-wise shrinkage coefficient. We compare the performance of our estimator with the SFS and the PSMC estimator through simulation studies.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/174276/1/yifanjin_1.pd