92,236 research outputs found
Complexity of Inference in Graphical Models
Graphical models provide a convenient representation for a broad class of probability distributions.
Due to their powerful and sophisticated modeling capabilities, such models have
found numerous applications in machine learning and other areas. In this paper we consider the
complexity of commonly encountered tasks involving graphical models such as the computation
of the mode of a posterior probability distribution (i.e., MAP estimation), and the computation
of marginal probabilities or the partition function. It is well-known that such inference problems
are hard in the worst case, but are tractable for models with bounded treewidth. We ask
whether treewidth is the only structural criterion of the underlying graph that enables tractable
inference. In other words, is there some class of structures with unbounded treewidth in which
inference is tractable? Subject to a combinatorial hypothesis due to Robertson, Seymour, and
Thomas (1994), we show that low treewidth is indeed the only structural restriction that can
ensure tractability. More precisely we show that for every growing family of graphs indexed
by tree-width, there exists a choice of potential functions such that the corresponding inference
problem is intractable. Thus even for the "best case" graph structures of high treewidth, there is
no polynomial-time inference algorithm. Our analysis employs various concepts from complexity theory and graph theory, with graph minors playing a prominent role
Block Belief Propagation for Parameter Learning in Markov Random Fields
Traditional learning methods for training Markov random fields require doing
inference over all variables to compute the likelihood gradient. The iteration
complexity for those methods therefore scales with the size of the graphical
models. In this paper, we propose \emph{block belief propagation learning}
(BBPL), which uses block-coordinate updates of approximate marginals to compute
approximate gradients, removing the need to compute inference on the entire
graphical model. Thus, the iteration complexity of BBPL does not scale with the
size of the graphs. We prove that the method converges to the same solution as
that obtained by using full inference per iteration, despite these
approximations, and we empirically demonstrate its scalability improvements
over standard training methods.Comment: Accepted to AAAI 201
Recent advances in imprecise-probabilistic graphical models
We summarise and provide pointers to recent advances in inference and identification for specific types of probabilistic graphical models using imprecise probabilities. Robust inferences can be made in so-called credal networks when the local models attached to their nodes are imprecisely specified as conditional lower previsions, by using exact algorithms whose complexity is comparable to that for the precise-probabilistic counterparts
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