Posterior distributions arising out of coalescent-based models of genetic diversity are nearly always intractable. Markov chain Monte Carlo methods, typically based on the Metropolis-Hastings algorithm, are gold-standard tools for sampling from these posteriors. However, their computational cost scales notoriously badly with problem complexity, so that their practical use is restricted to data sets which are small by modern standards. Over the past 5 years, several classes of nonreversible MCMC methods which use the gradient of the posterior density to guide chains during runs have become increasingly prominent as competitors of Metropolis-Hastings. These methods have better theoretical scaling properties than Metropolis-Hastings, but cannot be readily implemented for coalescent models because a posterior defined on discrete tree topologies does not have a natural analogue of a gradient. I will demonstrate how embedding spaces of coalescent trees into a continuous space facilitates the use of gradient information in algorithm specification, and also provides a framework for designing adaptive MCMC algorithms in a principled way. I will also outline how the same approach can be generalised to other state spaces involving boundaries, discrete variables, or disconnected regions. Comparisons with Metropolis-Hastings on simple coalescent examples show that these methods speed up mixing over the posterior, sometimes dramatically.
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Jere Koskela - holds an MMORSE, a MSc and a PhD from the University of Warwick (UK), where he is currently an Associate Professor at the Department of Statistics. He was also a postdoctoral researcher at Institut für Mathematik, Technische Universität Berlin, Germany. His research interests include Monte Carlo methods, statistical inference from stochastic processes and in settings with intractable likelihood, Bayesian nonparametric statistics, coalescent processes, and mathematical population genetics.