Wasserstein-Splitting Gaussian Process Regression for Heterogeneous Online Bayesian Inference

Gaussian processes (GPs) are a well-known nonparametric Bayesian inference technique, but they suffer from scalability problems for large sample sizes, and their performance can degrade for non-stationary or spatially heterogeneous data. In this work, we seek to overcome these issues through (i) employing variational free energy approximations of GPs operating in tandem with online expectation propagation steps; and (ii) introducing a local splitting step which instantiates a new GP whenever the posterior distribution changes significantly as quantified by the Wasserstein metric over posterior distributions. Over time, then, this yields an ensemble of sparse GPs which may be updated incrementally, and adapts to locality, heterogeneity, and non-stationarity in training data. We provide a 1-dimensional example to illustrate the motivation behind our approach, and compare the performance of our approach to other Gaussian process methods across various data sets, which often achieves competitive, if not superior predictive performance, relative to other locality-based GP regression methods in which hyperparameters are learned in an online manner.