  We study the Gaussian Process regression model in the context of training
data with noise in both input and output. The presence of two sources of noise
makes the task of learning accurate predictive models extremely challenging.
However, in some instances additional constraints may be available that can
reduce the uncertainty in the resulting predictive models. In particular, we
consider the case of monotonically ordered latent input, which occurs in many
application domains that deal with temporal data. We present a novel inference
and learning approach based on non-parametric Gaussian variational
approximation to learn the GP model while taking into account the new
constraints. The resulting strategy allows one to gain access to posterior
estimates of both the input and the output and results in improved predictive
performance. We compare our proposed models to state-of-the-art Noisy Input
Gaussian Process (NIGP) and other competing approaches on synthetic and real
sea-level rise data. Experimental results suggest that the proposed approach
consistently outperforms selected methods while, at the same time, reducing the
computational costs of learning and inference.
