The Multivariate Generalised von Mises Distribution: Inference and Applications

Alexandre Khae Wu Navarro, Jes Frellsen, Richard Turner

Research output: Conference Article in Proceeding or Book/Report chapterArticle in proceedingsResearchpeer-review


Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First we introduce a new multivariate distribution over circular variables, called the multivariate Generalised von Mises (mGvM) distribution. This distribution can be constructed by restricting and renormalising a general multivariate Gaussian distribution to the unit hyper-torus. Previously proposed multivariate circular distributions are shown to be special cases of this construction. Second, we introduce a new probabilistic model for circular regression inspired by Gaussian Processes, and a method for probabilistic Principal Component Analysis with circular hidden variables. These models can leverage standard modelling tools (e.g. kernel functions and automatic relevance determination). Third, we show that the posterior distribution in these models is a mGvM distribution which enables development of an efficient variational free-energy scheme for performing approximate inference and approximate maximum-likelihood learning.
Original languageEnglish
Title of host publicationProceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17)
PublisherAAAI Press
Publication date2017
ISBN (Print)N/A
Publication statusPublished - 2017


  • Circular variables
  • Probabilistic modelling
  • Multivariate Generalised von Mises distribution
  • Circular regression
  • Probabilistic Principal Component Analysis (PCA)


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