Exact and approximate inference in graphical models: variable elimination and beyond

Probabilistic graphical models offer a powerful framework to account for the dependence structure between variables, which is represented as a graph. However, the dependence between variables may render inference tasks intractable. In this paper we review techniques exploiting the graph structure for exact inference, borrowed from optimisation and computer science. They are built on the principle of variable elimination whose complexity is dictated in an intricate way by the order in which variables are eliminated. The so-called treewidth of the graph characterises this algorithmic complexity: low-treewidth graphs can be processed efficiently. The first message that we illustrate is therefore the idea that for inference in graphical model, the number of variables is not the limiting factor, and it is worth checking for the treewidth before turning to approximate methods. We show how algorithms providing an upper bound of the treewidth can be exploited to derive a 'good' elimination order enabling to perform exact inference. The second message is that when the treewidth is too large, algorithms for approximate inference linked to the principle of variable elimination, such as loopy belief propagation and variational approaches, can lead to accurate results while being much less time consuming than Monte-Carlo approaches. We illustrate the techniques reviewed in this article on benchmarks of inference problems in genetic linkage analysis and computer vision, as well as on hidden variables restoration in coupled Hidden Markov Models.