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We undertake Bayesian learning of the high-dimensional functionalrelationship between a system parameter vector and an observable, that is ingeneral tensor-valued. The ultimate aim is Bayesian inverse prediction of thesystem parameters, at which test data is recorded. We attempt such learninggiven hypercuboidally-shaped data that displays strong discontinuities,rendering learning challenging. We model the sought high-dimensional function,with a tensor-variate Gaussian Process (GP), and use three independent ways forlearning covariance matrices of the resulting likelihood, which isTensor-Normal. We demonstrate that the discontinuous data demands thatimplemented covariance kernels be non-stationary--achieved by modelling eachkernel hyperparameter, as a function of the sample function of the invokedtensor-variate GP. Each such function can be shown to be temporally-evolving,and treated as a realisation from a distinct scalar-variate GP, with covariancedescribed adaptively by collating information from a historical set of samplesof chosen sample-size. We prove that deep-learning using 2-"layers", suffice,where the outer-layer comprises the tensor-variate GP, compounded with multiplescalar-variate GPs in the "inner-layer", and undertake inference withMetropolis-within-Gibbs. We apply our method to a cuboidally-shaped,discontinuous, real dataset, and subsequently perform forward prediction togenerate data from our model, given our results--to perform model-checking.

Deep Bayesian Supervised Learning given Hypercuboidally-shaped,  Discontinuous Data, using Compound Tensor-Variate & Scalar-Variate Gaussian  Processes