A New Reliable & Parsimonious Learning Strategy Comprising Two Layers of Gaussian Processes, to Address Inhomogeneous Empirical Correlation Structures

We present a new strategy for learning the functional relation between a pair of variables, while addressing inhomogeneities in the correlation structure of the available data, by modelling the sought function as a sample function of a non-stationary Gaussian Process (GP), that nests within itself multiple other GPs, each of which we prove can be stationary, thereby establishing sufficiency of two GP layers. In fact, a non-stationary kernel is envisaged, with each hyperparameter set as dependent on the sample function drawn from the outer non-stationary GP, such that a new sample function is drawn at every pair of input values at which the kernel is computed. However, such a model cannot be implemented, and we substitute this by recalling that the average effect of drawing different sample functions from a given GP is equivalent to that of drawing a sample function from each of a set of GPs that are rendered different, as updated during the equilibrium stage of the undertaken inference (via MCMC). The kernel is fully non-parametric, and it suffices to learn one hyperparameter per layer of GP, for each dimension of the input variable. We illustrate this new learning strategy on a real dataset.