Automorphism Groups of Graphical Models and Lifted Variational Inference

Classical approaches to probabilistic inference - an area now reasonably well understood - have traditionally exploited low tree-width and sparsity of the graphical model for efficient exact and approximate inference. A more recent approach known as lifted inference [2, 12, 6, 7] has demonstrated the possibility to perform very efficient inference in highly-connected, but symmetric models such as those arising in the context of relational (or first-order) probabilistic models. While it is clear that symmetry is the essential element in lifted inference, there is currently no formally defined notion of symmetry of a probabilistic model, and thus no formal account of what "exploiting symmetry" means in lifted inference. The mathematical formulation of symmetry of an object is typically defined via a set of transformations that preserve the object of interest. Since this set forms a mathematical group (so-called the automorphism group of that object), the theory of groups and group action are essential in the study of symmetry. In this paper, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the notion of symmetry of a general graphical model. This automorphism group provides a precise mathematical framework for lifted inference in graphical models.