  Probabilistic representations, such as Bayesian and Markov networks, are
fundamental to much of statistical machine learning. Thus, learning
probabilistic representations directly from data is a deep challenge, the main
computational bottleneck being inference that is intractable. Tractable
learning is a powerful new paradigm that attempts to learn distributions that
support efficient probabilistic querying. By leveraging local structure,
representations such as sum-product networks (SPNs) can capture high tree-width
models with many hidden layers, essentially a deep architecture, while still
admitting a range of probabilistic queries to be computable in time polynomial
in the network size. The leaf nodes in SPNs, from which more intricate mixtures
are formed, are tractable univariate distributions, and so the literature has
focused on Bernoulli and Gaussian random variables. This is clearly a
restriction for handling mixed discrete-continuous data, especially if the
continuous features are generated from non-parametric and non-Gaussian
distribution families. In this work, we present a framework that systematically
integrates SPN structure learning with weighted model integration, a recently
introduced computational abstraction for performing inference in hybrid
domains, by means of piecewise polynomial approximations of density functions
of arbitrary shape. Our framework is instantiated by exploiting the notion of
propositional abstractions, thus minimally interfering with the SPN structure
learning module, and supports a powerful query interface for conditioning on
interval constraints. Our empirical results show that our approach is
effective, and allows a study of the trade off between the granularity of the
learned model and its predictive power.
