  The field of statistical relational learning aims at unifying logic and
probability to reason and learn from data. Perhaps the most successful paradigm
in the field is probabilistic logic programming: the enabling of stochastic
primitives in logic programming, which is now increasingly seen to provide a
declarative background to complex machine learning applications. While many
systems offer inference capabilities, the more significant challenge is that of
learning meaningful and interpretable symbolic representations from data. In
that regard, inductive logic programming and related techniques have paved much
of the way for the last few decades.
  Unfortunately, a major limitation of this exciting landscape is that much of
the work is limited to finite-domain discrete probability distributions.
Recently, a handful of systems have been extended to represent and perform
inference with continuous distributions. The problem, of course, is that
classical solutions for inference are either restricted to well-known
parametric families (e.g., Gaussians) or resort to sampling strategies that
provide correct answers only in the limit. When it comes to learning, moreover,
inducing representations remains entirely open, other than "data-fitting"
solutions that force-fit points to aforementioned parametric families.
  In this paper, we take the first steps towards inducing probabilistic logic
programs for continuous and mixed discrete-continuous data, without being
pigeon-holed to a fixed set of distribution families. Our key insight is to
leverage techniques from piecewise polynomial function approximation theory,
yielding a principled way to learn and compositionally construct density
functions. We test the framework and discuss the learned representations.
