  Using the theory of group action, we first introduce the concept of the
automorphism group of an exponential family or a graphical model, thus
formalizing the general notion of symmetry of a probabilistic model. This
automorphism group provides a precise mathematical framework for lifted
inference in the general exponential family. Its group action partitions the
set of random variables and feature functions into equivalent classes (called
orbits) having identical marginals and expectations. Then the inference problem
is effectively reduced to that of computing marginals or expectations for each
class, thus avoiding the need to deal with each individual variable or feature.
We demonstrate the usefulness of this general framework in lifting two classes
of variational approximation for maximum a posteriori (MAP) inference: local
linear programming (LP) relaxation and local LP relaxation with cycle
constraints; the latter yields the first lifted variational inference algorithm
that operates on a bound tighter than the local constraints.
