  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, that is 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. covariance functions
and methods for 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.
