Inference of Factor Graphs under Topological Transformations

Graphical models and factor graphs are probabilistic models that incorporate prior knowledge of dependencies between variables; celebrated examples include hidden Markov models, and higher-order interactions are accounted for through hypergraphs. Computing the posterior distribution for a given collection of observations is called inference and is, in general, computationally very costly. In practice, one often resorts to variational inference, which consists in optimizing a weighted sum free energy over subcollections of variables, under the constraint that their probability distributions are compatible by marginalization. This compatibility condition defines the space of sections of specific presheaves, which we call projection presheaves. The General Belief Propagation algorithm is used to find the critical points of the weighted free energy. We will first explain how one can extend factor graphs to account for a broader class of relations between subcollections of variab!

les, by generalizing results from those specific presheaves to arbitrary presheaves over a poset. Given this broader framework, we show that the algorithm is functorial with respect to natural transformations. We then show that inference on minimal deformation retracts of hypergraphs seen as posets is sufficient for inference on the entire poset, and that for projection presheaves, a similar result holds when considering the undirected graph underlying the hypergraph. In collaboration with Léo Boitel.