  A novel multi-task Gaussian process (GP) framework is proposed, by using a
common mean process for sharing information across tasks. In particular, we
investigate the problem of time series forecasting, with the objective to
improve multiple-step-ahead predictions. The common mean process is defined as
a GP for which the hyper-posterior distribution is tractable. Therefore an EM
algorithm is derived for handling both hyper-parameters optimisation and
hyper-posterior computation. Unlike previous approaches in the literature, the
model fully accounts for uncertainty and can handle irregular grids of
observations while maintaining explicit formulations, by modelling the mean
process in a unified GP framework. Predictive analytical equations are
provided, integrating information shared across tasks through a relevant prior
mean. This approach greatly improves the predictive performances, even far from
observations, and may reduce significantly the computational complexity
compared to traditional multi-task GP models. Our overall algorithm is called
\textsc{Magma} (standing for Multi tAsk Gaussian processes with common MeAn).
The quality of the mean process estimation, predictive performances, and
comparisons to alternatives are assessed in various simulated scenarios and on
real datasets.
