
Negative Tree Reweighted Belief Propagation
We introduce a new class of lower bounds on the log partition function o...
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Piecewise Training for Undirected Models
For many large undirected models that arise in realworld applications, ...
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Fixed Points of Belief Propagation  An Analysis via Polynomial Homotopy Continuation
Belief propagation (BP) is an iterative method to perform approximate in...
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Approximate Inference and Constrained Optimization
Loopy and generalized belief propagation are popular algorithms for appr...
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Upper Bounds via Lamination on the Constrained Secrecy Capacity of Hypergraphical Sources
Hypergraphical sources are a natural class of sources for secret key gen...
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On the Partition Bound for Undirected Unicast Network Information Capacity
One of the important unsolved problems in information theory is the conj...
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What Cannot be Learned with Bethe Approximations
We address the problem of learning the parameters in graphical models wh...
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A New Class of Upper Bounds on the Log Partition Function
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, that is applicable to an arbitrary undirected graphical model. In the special case of convex combinations of treestructured distributions, we obtain a family of variational problems, similar to the Bethe free energy, but distinguished by the following desirable properties: i. they are cnvex, and have a unique global minimum; and ii. the global minimum gives an upper bound on the log partition function. The global minimum is defined by stationary conditions very similar to those defining fixed points of belief propagation or treebased reparameterization Wainwright et al., 2001. As with BP fixed points, the elements of the minimizing argument can be used as approximations to the marginals of the original model. The analysis described here can be extended to structures of higher treewidth e.g., hypertrees, thereby making connections with more advanced approximations e.g., Kikuchi and variants Yedidia et al., 2001; Minka, 2001.
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