Speaker: Dr. Ben Calderhead, Department of Mathematics, Imperial College London
Title: Quasi Markov Chain Monte Carlo Methods

Abstract: Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior densities within the Bayesian framework, in particular for inverse problems. We introduce a general parallel Markov chain Monte Carlo(MCMC) framework, for which we prove a law of large numbers and a central limit theorem. In that context, non-reversible transitions are investigated. We then extend this approach to the use of adaptive kernels and state conditions, under which ergodicity holds. As a further extension, an importance sampling estimator is derived, for which asymptotic unbiasedness is proven. We consider the use of completely uniformly distributed (CUD) numbers within the above mentioned algorithms, which leads to a general parallel quasi-MCMC (QMCMC) methodology. We prove consistency of the resulting estimators and demonstrate numerically that this approach scales close to n^{-2} as we increase parallelisation, instead of the usual n^{-1} that is typical of standard MCMC algorithms. In practical statistical models we observe multiple orders of magnitude improvement compared with pseudo-random methods.

Prof. Renauld Lambiote, University of Oxford, UK (15:00-16:00)

Higher-Order Networks:

Network science provides powerful analytical and computational methods to describe the behaviour of complex systems. From a networks viewpoint, the system is seen as a collection of elements interacting through pairwise connections. Canonical examples include social networks, neuronal networks or the Web. Importantly, elements often interact directly with a relatively small number of other elements, while they may influence large parts of the system indirectly via chains of direct interactions. In other words, networks allow for a sparse architecture together with global connectivity. Compared with mean-field approaches, network models often have greater explanatory power because they account for the non-random topologies of real-life systems. However, new forms of high-dimensional and time-resolved data have now also shed light on the limitations of these models. In this talk, I will review recent advances in the development of higher-order network models, which account for different types of higher-order dependencies in complex data. Those include temporal networks, where the network is itself a dynamical entity and higher-order Markov models, where chains of interactions are more than a combination of links.

Presented by: Alison Etheridge OBE FRS - University of Oxford

When Mendelian genetics was rediscovered at the beginning of the 20th Century, it was widely believed to be incompatible with Darwin's theory of natural selection. The mathematical sciences, in the hands of pioneers such as Fisher, Haldane and Wright, played a fundamental role in the reconciliation of the two theories, and the new field of theoretical population genetics was born. But fundamental questions remained (and remain) unresolved. The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns that we see today? Whereas the pioneers of the field could only observe genetic variation indirectly, by looking at traits of individuals in a population, researchers today have direct access to DNA sequences, but making sense of this wealth of data presents a major scientific challenge and mathematical models play a decisive role. In this lecture we'll discuss how to distill our understanding into workable models and then explore the remarkable power of simple mathematical caricatures in interrogating modern genetic data.

Free attendance

There will be a reception after the lecture

Main contact point: paula.matthews@warwick.ac.uk

Dr. Yoav Zemel, University of Göttingen, Germany (15:00-16:00)

Title: Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Abstract: Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality of these operators. We describe the manifold-like geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric (Pigoli et al. Biometrika 101, 409–422, 2014). We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a detailed description of those aspects of this manifold-like geometry that are important in terms of statistical inference; to establish key properties of the Fréchet mean of a random sample of covariances; and to define generative models that are canonical for such metrics and link with the problem of registration of warped functional data.