  When modelling censored observations, a typical approach in current
regression methods is to use a censored-Gaussian (i.e. Tobit) model to describe
the conditional output distribution. In this paper, as in the case of missing
data, we argue that exploiting correlations between multiple outputs can enable
models to better address the bias introduced by censored data. To do so, we
introduce a heteroscedastic multi-output Gaussian process model which combines
the non-parametric flexibility of GPs with the ability to leverage information
from correlated outputs under input-dependent noise conditions. To address the
resulting inference intractability, we further devise a variational bound to
the marginal log-likelihood suitable for stochastic optimization. We
empirically evaluate our model against other generative models for censored
data on both synthetic and real world tasks and further show how it can be
generalized to deal with arbitrary likelihood functions. Results show how the
added flexibility allows our model to better estimate the underlying
non-censored (i.e. true) process under potentially complex censoring dynamics.
