Marginal analysis on binary pairwise Gibbs random fields

In this paper, we study marginal problems for a class of binary pairwise Gibbs random fields (BPW-GRFs). Given a BPW-GRF associated with a family of binary positive pairwise potentials, finding the exact marginal for each random variable is typically an NP-hard problem. In this paper, we develop upper and lower bounds of the true marginals in BPW-GRFs. Our bounds can be easily computed via an iteration on appropriate trees that are constructed from the corresponding BPW-GRF graphs. We prove that these marginal bounds outperform existing bounds. We also show via simulations that this improvement is significant on graphs with weak potentials.