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We thank all the reviewers for their time in giving us reviews and feedback. One point deserves specific comment: R1, R2, and R4 all had questions about the relationship between hypertree width and hierarchy width, and how this relates to the comparison between Gibbs sampling and exact inference techniques. When hierarchy width is bounded, the hypertree width is similarly bounded (Statement 1 in our paper). This means that for the models we focus on, where Gibbs mixes in polynomial time, exact inference also runs in polynomial time. However, for graphs with sufficiently small weights (such as the Paleontology model we mention), the polynomial exponent for Gibbs will be smaller than for exact sampling.