  A large class of modern probabilistic learning systems assumes symmetric
distributions, however, real-world data tend to obey skewed distributions and
are thus not always adequately modelled through symmetric distributions. To
address this issue, elliptical distributions are increasingly used to
generalise symmetric distributions, and further improvements to skewed
elliptical distributions have recently attracted much attention. However,
existing approaches are either hard to estimate or have complicated and
abstract representations. To this end, we propose to employ the
von-Mises-Fisher (vMF) distribution to obtain an explicit and simple
probability representation of the skewed elliptical distribution. This is shown
not only to allow us to deal with non-symmetric learning systems, but also to
provide a physically meaningful way of generalising skewed distributions. For
rigour, our extension is proved to share important and desirable properties
with its symmetric counterpart. We also demonstrate that the proposed vMF
distribution is both easy to generate and stable to estimate, both
theoretically and through examples.
