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Probabilistic Machine Learning for Circular Statistics: Models and inference using the Multivariate Generalised von Mises distribution 10.17863/CAM.26449 Wu Navarro, Alexandre Khae 0000000314528773 Turner, Richard Probabilistic machine learning and circular statistics—the branch of statistics concerned with data as angles and directions—are two research communities that have grown mostly in isolation from one another. On the one hand, probabilistic machine learning community has developed powerful frameworks for problems whose data lives on Euclidean spaces, such as Gaussian Processes, but have generally neglected other topologies studied by circular statistics. On the other hand, the approximate inference frameworks from probabilistic machine learning have only recently started to the circular statistics landscape. This thesis intends to redress the gap between these two fields by contributing to both fields with models and approximate inference algorithms. In particular, we introduce the multivariate Generalised von Mises distribution (mGvM), which allows the use of kernels in circular statistics akin to Gaussian Processes, and an augmented representation. These models account for a vast number of applications comprising both latent variable modelling and regression of circular data. Then, we propose methods to conduct approximate inference on these models. In particular, we investigate the use of Variational Inference, Expectation Propagation and Markov chain Monte Carlo methods. The variational inference route taken was a mean field approach to efficiently leverage the mGvM tractable conditionals and create a baseline for comparison with other methods. Then, an Expectation Propagation approach is presented drawing on the Expectation Consistent Framework for Ising models and connecting the approximations used to the augmented model presented. In the final MCMC chapter, efficient Gibbs and Hamiltonian Monte Carlo samplers are derived for the mGvM and the augmented model. University of Cambridge 2019-04-27 Thesis Doctoral Doctor of Philosophy (PhD) en Cambridge Trusts CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior https://www.repository.cam.ac.uk/handle/1810/279067 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a3cc7f6f-61d0-4151-be0e-e446bb6691ed/download b7c76c5a051e6df8da41e42c8e629586 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9e9e8501-0491-434d-b70f-beedd07fe4ef/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by-nc-nd/4.0/ Machine Learning Circular Statistics von Mises distribution Gaussian Processes Probabilistic models Approximate Inference