Heteroscedastic Gaussian processes for uncertain and incomplete data

In probabilistic inference, many implicit and explicit assumptions are taken about the nature of input noise and the function fit to either simplify the mathematics, improve the time complexity or optimise for space. It is often assumed that the inputs are noiseless or that the noise is drawn from the same distribution for all inputs, that all the variables used during training will be present during prediction and with the same degrees of uncertainties, and that the confidence about the prediction is uniform across the input space. This thesis presents a more generalised sparse Gaussian process model that relaxes these assumptions to inputs with variable degrees of uncertainty, or completeness in the input, and produces variable uncertainty estimation over the output. The capabilities of sparse Gaussian processes are further enhanced to allow for non-stationarity which minimises the number of required basis functions, a prior mean function for better extrapolation performance and cost-sensitive learning for non-uniform weighting of samples. The results are demonstrated on an astrophysical problem of estimating galactic redshifts from their photometry. This problem, by its nature, can capitalise on the features of the proposed model as the noise on the photometry can vary across different galaxies or catalogues, not all photometry might be available during prediction or shared amongst different surveys, and the input-dependent uncertainty estimation gives astrophysicists the ability to trade off completeness for accuracy to answer a range of different questions related to astronomy and cosmology.