  Recent research has made significant progress on the problem of bounding log
partition functions for exponential family graphical models. Such bounds have
associated dual parameters that are often used as heuristic estimates of the
marginal probabilities required in inference and learning. However these
variational estimates do not give rigorous bounds on marginal probabilities,
nor do they give estimates for probabilities of more general events than simple
marginals. In this paper we build on this recent work by deriving rigorous
upper and lower bounds on event probabilities for graphical models. Our
approach is based on the use of generalized Chernoff bounds to express bounds
on event probabilities in terms of convex optimization problems; these
optimization problems, in turn, require estimates of generalized log partition
functions. Simulations indicate that this technique can result in useful,
rigorous bounds to complement the heuristic variational estimates, with
comparable computational cost.
