  Incorporating constraints is a major concern in probabilistic machine
learning. A wide variety of problems require predictions to be integrated with
reasoning about constraints, from modelling routes on maps to approving loan
predictions. In the former, we may require the prediction model to respect the
presence of physical paths between the nodes on the map, and in the latter, we
may require that the prediction model respect fairness constraints that ensure
that outcomes are not subject to bias. Broadly speaking, constraints may be
probabilistic, logical or causal, but the overarching challenge is to determine
if and how a model can be learnt that handles all the declared constraints. To
the best of our knowledge, this is largely an open problem. In this paper, we
consider a mathematical inquiry on how the learning of tractable probabilistic
models, such as sum-product networks, is possible while incorporating
constraints.
