Theory and Methods of Data Science

Scalable inference for a full multivariate stochastic volatility model
Petros Dellaportas, Anastasios Plataniotis, Michalis Titsias
University College London
We introduce a multivariate stochastic volatility model for asset returns that imposes no restrictions to the structure of the volatility matrix and treats all its elements as functions of latent stochastic processes. When the number of assets is prohibitively large, we propose a factor multivariate stochastic volatility model in which the variances and correlations of the factors evolve stochastically over time. Inference is achieved via a carefully designed
feasible and scalable Markov chain Monte Carlo algorithm that combines two computationally important ingredients: it utilizes invariant to the prior Metropolis proposal densities for simultaneously updating all latent paths and has quadratic, rather than cubic, computational complexity when evaluating the multivariate normal densities required. We apply our modelling and computational methodology to 571 stock daily returns of Euro
STOXX index for data over a period of 10 years.

Topological Data Analysis at the ATI
Ulrike Tillmann
University of Oxford
The talk will introduce topological data analysis (TDA). The main goal is to give the audience a sense of this relatively new area of research, its challenges and its opportunities. Drawing on classical geometry and topology, TDA provides a flexible tool generating global statistics for data that has been applied to a large array of domains, including medicine, computer vision, and manufacturing. Time permitting we will touch on some of the research questions that will be persued at the Alan Turing Institute.

Bernstein-von Mises theorem and high dimensional nonidentifiable possibly misspecified models
Natalia Bochkina, Peter Green
University of Edinburgh
We consider a broad class of statistical models that can be misspecified and ill-posed, from a Bayesian perspective. This provides a flexible and interpretable framework for their analysis, but it is important to understand the relationship between the chosen Bayesian model and the resulting solution, especially in the ill-posed case where in the absence of prior information the solution is not unique. Compared to earlier work about the Bernstein-von Mises theorem for nonregular well-posed Bayesian models, we show that non-identifiable part of the likelihood, together with the constraints on the parameter space, introduce a more complex geometric structure of the posterior distribution around the best reconstruction point in the limit, and provide a local approximation of the posterior distribution in this neighbourhood. The results apply to misspecified models which allows, for instance, to evaluate the effect of of model approximation on statistical inference. Emission tomography is taken as a canonical example for study, but our results hold for a
wider class of generalised linear inverse problems with constraints.