  Aggregated data is commonplace in areas such as epidemiology and demography.
For example, census data for a population is usually given as averages defined
over time periods or spatial resolutions (cities, regions or countries). In
this paper, we present a novel multi-task learning model based on Gaussian
processes for joint learning of variables that have been aggregated at
different input scales. Our model represents each task as the linear
combination of the realizations of latent processes that are integrated at a
different scale per task. We are then able to compute the cross-covariance
between the different tasks either analytically or numerically. We also allow
each task to have a potentially different likelihood model and provide a
variational lower bound that can be optimised in a stochastic fashion making
our model suitable for larger datasets. We show examples of the model in a
synthetic example, a fertility dataset, and an air pollution prediction
application.
