Reviews: Synthesis of MCMC and Belief Propagation

The authors develop an very interesting method to the hard problem of computing the partition function of GM. In particular, there is no method to compute Z for general graphs accurately and log(Z) is a particularly interesting component in many fields: it is related to the likelihood in Machine Learning, it gives all the statistical information of a system in statistical mechanic, ... I think therefore that any effort toward that direction is an important contribution. Concerning the work presented here, I have some questions and concerns that I would like the authors to answer: 1) the authors mentioned that BP in general fail to converge on general graph but that more involved convergent alternative exists. I think the author should be more specific, precising if all the convergent algorithms do provide the same estimation of the BP partition function and it they are indeed linear in the system size. The 2-reg loop can be proven to converge in polynomial time but the full loop does not have these garranty.