An approximate inference with Gaussian process to latent functions from uncertain data

Most formulations of supervised learning are often based on the assumption that only the outputs data are uncertain. However, this assumption might be too strong for some learning tasks. This paper investigates the use of Gaussian processes to infer latent functions from a set of uncertain input-output examples. By assuming Gaussian distributions with known variances over the inputs-outputs and using the expectation of the covariance function, it is possible to analytically compute the expected covariance matrix of the data to obtain a posterior distribution over functions. The method is evaluated on a synthetic problem and on a more realistic one, which consist in learning the dynamics of a cart-pole balancing task. The results indicate an improvement of the mean squared error and the likelihood of the posterior Gaussian process when the data uncertainty is significant.