A Convex Upper Bound on the Log-Partition Function for Binary Graphical Models

We consider the problem of bounding from above the log-partition function corresponding to second-order Ising models for binary distributions. We introduce a new bound, the cardinality bound, which can be computed via convex optimization. The corresponding error on the logpartition function is bounded above by twice the distance, in model parameter space, to a class of “standard” Ising models, for which variable inter-dependence is described via a simple mean field term. In the context of maximum-likelihood, using the new bound instead of the exact log-partition function, while constraining the distance to the class of standard Ising models, leads not only to a good approximation to the log-partition function, but also to a model that is parsimonious, and easily interpretable. We compare our bound with the log-determinant bound introduced by Wainwright and Jordan (2006), and show that when the l1-norm of the model parameter vector is small enough, the latter is outperformed by the new bound. ∗Research supported in part by NSF grant DMS-0625371.