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Tensor-valued data are being encountered increasingly more commonly, in thebiological, natural as well as the social sciences. The learning of the unknownmodel parameter vector given such data, involves covariance modelling of suchdata, though this can be difficult owing to the high-dimensional nature of thedata, where the numerical challenge of such modelling can only be compounded bythe largeness of the available data set. Assuming such data to be modelledusing a correspondingly high-dimensional Gaussian Process (${\cal  GP}$), the joint density of a finite set of such data sets is then a tensornormal distribution, with density parametrised by a mean tensor$\boldsymbol{M}$ (that is of the same dimensionality as the $k$-tensor valuedobservable), and the $k$ covariance matrices$\boldsymbol{\Sigma}_1,...,\boldsymbol{\Sigma}_k$. When aiming to model thecovariance 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 covariancemodelling by first expressing the probability density of the available datasets as tensor-normal. We then invoke appropriate priors on these unknownparameters and express the posterior of the unknowns given the data. We samplefrom this posterior using an appropriate variant of Metropolis Hastings. Sincethe classical MCMC is time and resource intensive in high-dimensional statespaces, we use an efficient variant of the Metropolis-Hastingsalgorithm--Transformation based MCMC--employed to perform efficient samplingfrom a high-dimensional state space. Once we perform the covariance modellingof such a data set, we will learn the unknown model parameter vector at which ameasured (or test) data set has been obtained, given the already modelled data(training data), augmented by the test data.

Bayesian Covariance Modelling of Large Tensor-Variate Data Sets $\&$ Inverse Non-parametric Learning of the Unknown Model Parameter Vector