Scaling up Hybrid Probabilistic Inference with Logical and Arithmetic
  Constraints via Message Passing
  Weighted model integration (WMI) is a very appealing framework for
probabilistic inference: it allows to express the complex dependencies of
real-world problems where variables are both continuous and discrete, via the
language of Satisfiability Modulo Theories (SMT), as well as to compute
probabilistic queries with complex logical and arithmetic constraints. Yet,
existing WMI solvers are not ready to scale to these problems. They either
ignore the intrinsic dependency structure of the problem at all, or they are
limited to too restrictive structures. To narrow this gap, we derive a
factorized formalism of WMI enabling us to devise a scalable WMI solver based
on message passing, MP-WMI. Namely, MP-WMI is the first WMI solver which allows
to: 1) perform exact inference on the full class of tree-structured WMI
problems; 2) compute all marginal densities in linear time; 3) amortize
inference inter query. Experimental results show that our solver dramatically
outperforms the existing WMI solvers on a large set of benchmarks.
