  Tensor-valued data are being encountered increasingly more commonly, in the
biological, natural as well as the social sciences. The learning of the unknown
model parameter vector given such data, involves covariance modelling of such
data, though this can be difficult owing to the high-dimensional nature of the
data, where the numerical challenge of such modelling can only be compounded by
the largeness of the available data set. Assuming such data to be modelled
using a correspondingly high-dimensional Gaussian Process (${\cal
  GP}$), the joint density of a finite set of such data sets is then a tensor
normal distribution, with density parametrised by a mean tensor
$\boldsymbol{M}$ (that is of the same dimensionality as the $k$-tensor valued
observable), and the $k$ covariance matrices
$\boldsymbol{\Sigma}_1,...,\boldsymbol{\Sigma}_k$. When aiming to model the
covariance structure of the data, we need to estimate/learn
$\{\boldsymbol{\Sigma}_1,...,\boldsymbol{\Sigma}_k \}$ and $\boldsymbol{M}$,
given tha data. We present a new method in which we perform such covariance
modelling by first expressing the probability density of the available data
sets as tensor-normal. We then invoke appropriate priors on these unknown
parameters and express the posterior of the unknowns given the data. We sample
from this posterior using an appropriate variant of Metropolis Hastings. Since
the classical MCMC is time and resource intensive in high-dimensional state
spaces, we use an efficient variant of the Metropolis-Hastings
algorithm--Transformation based MCMC--employed to perform efficient sampling
from a high-dimensional state space. Once we perform the covariance modelling
of such a data set, we will learn the unknown model parameter vector at which a
measured (or test) data set has been obtained, given the already modelled data
(training data), augmented by the test data.
