  Multitask Gaussian processes (MTGP) are the Gaussian process (GP) framework's
solution for multioutput regression problems in which the $T$ elements of the
regressors cannot be considered conditionally independent given the
observations. Standard MTGP models assume that there exist both a multitask
covariance matrix as a function of an intertask matrix, and a noise covariance
matrix. These matrices need to be approximated by a low rank simplification of
order $P$ in order to reduce the number of parameters to be learnt from $T^2$
to $TP$. Here we introduce a novel approach that simplifies the multitask
learning by reducing it to a set of conditioned univariate GPs without the need
for any low rank approximations, therefore completely eliminating the
requirement to select an adequate value for hyperparameter $P$. At the same
time, by extending this approach with both a hierarchical and an approximate
model, the proposed extensions are capable of recovering the multitask
covariance and noise matrices after learning only $2T$ parameters, avoiding the
validation of any model hyperparameter and reducing the overall complexity of
the model as well as the risk of overfitting. Experimental results over
synthetic and real problems confirm the advantages of this inference approach
in its ability to accurately recover the original noise and signal matrices, as
well as the achieved performance improvement in comparison to other state of
art MTGP approaches. We have also integrated the model with standard GP
toolboxes, showing that it is computationally competitive with state of the art
options.
