  Real-world signals typically span across multiple dimensions, that is, they
naturally reside on multi-way data structures referred to as tensors. In
contrast to standard ``flat-view'' multivariate matrix models which are
agnostic to data structure and only describe linear pairwise relationships, we
introduce the tensor-valued Gaussian distribution which caters for multilinear
interactions -- the linear relationship between fibers -- which is reflected by
the Kronecker separable structure of the mean and covariance. By virtue of the
statistical identifiability of the proposed distribution formulation, whereby
different parameter values strictly generate different probability
distributions, it is shown that the corresponding likelihood function can be
maximised analytically to yield the maximum likelihood estimator. For rigour,
the statistical consistency of the estimator is also demonstrated through
numerical simulations. The probabilistic framework is then generalised to
describe the joint distribution of multiple tensor-valued random variables,
whereby the associated mean and covariance exhibit a Khatri-Rao separable
structure. The proposed models are shown to serve as a natural basis for
gridded atmospheric climate modelling.
