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This paper presents a new approach to sampling from binary graphical models. There are two main tricks in this paper. The first is a clever construction to transform an arbitrary distribution over binary vectors x into a uniform distribution over an expanded space. This is done by augmenting x with additional bits, which are subject to parity constraints that ensure that the number of is approximately proportional to the original probability of x. This is a uniform distribution over a set with complex nonlinear constraints, so sampling from this distribution is challenging.