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We consider log-supermodular models on binary variables, which areprobabilistic models with negative log-densities which are submodular. Thesemodels provide probabilistic interpretations of common combinatorialoptimization tasks such as image segmentation. In this paper, we focusprimarily on parameter estimation in the models from known upper-bounds on theintractable log-partition function. We show that the bound based on separableoptimization on the base polytope of the submodular function is always inferiorto a bound based on "perturb-and-MAP" ideas. Then, to learn parameters, giventhat our approximation of the log-partition function is an expectation (overour own randomization), we use a stochastic subgradient technique to maximize alower-bound on the log-likelihood. This can also be extended to conditionalmaximum likelihood. We illustrate our new results in a set of experiments inbinary image denoising, where we highlight the flexibility of a probabilisticmodel to learn with missing data.

Parameter Learning for Log-supermodular Distributions