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Thanks to all 6 reviewers for your helpful comments, we try to address all points raised in what follows: To Reviewer 1, * thank you for the positive feedback, as for your last comment regarding theoretical guarantees, this is something we are currently looking into considering, among other things, "convergent EP" of Heskes and Zoeter, 2002. To Reviewer 2, * re: "better explanation of how to find the proposal" the proposal at each node at a given step is the current approximation of the belief on that node. I.e.: the product of the approximated messages coming into that node, this is explained in line 196-209 (page 4), in particular equation 14 gives the form of the proposals. This is one of the main point of our submission i.e., suggesting to construct a proposal on the go using EP. We could try to make sure this point is clearer.