  Multi-output regression seeks to borrow strength and leverage commonalities
across different but related outputs in order to enhance learning and
prediction accuracy. A fundamental assumption is that the output/group
membership labels for all observations are known. This assumption is often
violated in real applications. For instance, in healthcare datasets, sensitive
attributes such as ethnicity are often missing or unreported. To this end, we
introduce a weakly-supervised multi-output model based on dependent Gaussian
processes. Our approach is able to leverage data without complete group labels
or possibly only prior belief on group memberships to enhance accuracy across
all outputs. Through intensive simulations and case studies on an Insulin,
Testosterone and Bodyfat dataset, we show that our model excels in multi-output
settings with missing labels, while being competitive in traditional fully
labeled settings. We end by highlighting the possible use of our approach in
fair inference and sequential decision-making.
