  A variety of lifted inference algorithms, which exploit model symmetry to
reduce computational cost, have been proposed to render inference tractable in
probabilistic relational models. Most existing lifted inference algorithms
operate only over discrete domains or continuous domains with restricted
potential functions, e.g., Gaussian. We investigate two approximate lifted
variational approaches that are applicable to hybrid domains and expressive
enough to capture multi-modality. We demonstrate that the proposed variational
methods are both scalable and can take advantage of approximate model
symmetries, even in the presence of a large amount of continuous evidence. We
demonstrate that our approach compares favorably against existing
message-passing based approaches in a variety of settings. Finally, we present
a sufficient condition for the Bethe approximation to yield a non-trivial
estimate over the marginal polytope.
