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Response to reviewers Dear reviewers, Thank you all very much for your time and your comments on how to improve our presentation of our results. A common theme from your comments is a concern on whether our results and our methods are of general interest. We agree that the assumptions we use are restrictive. However, the proof technique we introduce is flexible and will extend to a more general setup (at the cost of heavier notation and longer calculations). The easiest extension is to suppose only that hybrid distributions are strongly log-concave, which is much less stringent than making assumptions on likelihood sites, and covers many latent Gaussian models, like logistic or probit regression.