Reviewers 1 and 2: We thank you for your time and feedback.  We will incorporate your suggestions into the final versions (including the interesting connection to dimension-free and infinite-dimensional MCMC).  Reviewer 1's point about the nature of the resulting flows is well-taken.  Indeed, the flows are not Hamiltonian in the sense that they generally do not solve Hamilton's equations in the original space; we will clarify the distinction in the final version.$
Reviewer 3: Respectfully, we assert that these concerns are based on a major misunderstanding of the HSS method. The review states, "The algorithm replaces the leapfrog moves in the original HMC by a slice sampling on the trajectory." We stress that this is not what HSS does. Whereas HMC uses the dynamics induced by the *posterior*, HSS uses the dynamics induced by the *prior* (or a transformation of the prior) to generate a trajectory on which to slice sample. This distinction is absolutely critical to understanding the difference between HMC and HSS. In many commonly used hierarchical models, a transformation of the prior (via the CDF) yields an analytical solution to Hamilton's equations, making the assumptions of HSS rather easy to satisfy, but certainly the posterior HMC trajectories are not in general analytical. Furthermore, the practitioner has the freedom to specify the prior so that HSS may be used; the strategy of specifying priors to allow a specific inference algorithm is common in practice (e.g., conjugate priors in Gibbs sampling and variational inference). Therefore HSS adds minimal restrictions, and the stated concern seems not to be relevant here.

We do appreciate the time and effort and will update the text of the paper, in particular the framing of the method, to make more clear this essential distinction between HMC and HSS.

For completeness, we also believe the other concerns in this review all appear to stem from the same principal misunderstanding. Specifically:

- "The proposed algorithm is only possible when the full analytical solution is available." HSS does not require the full analytical solution to the posterior Hamiltonian system; it merely requires the analytical solution to the prior (or transformed prior) Hamiltonian system. We agree that most Hamiltonian systems induced by the posterior are intractable (and that a method requiring analytical posterior trajectories would indeed be of limited use!), and that in fact is one motivation for HSS.

- "... if the model parameters could be decomposed as collections of conditionally independent variables, then applying the old-fashioned slice sampling would be adequate to finish all the work gracefully and efficiently." Although univariate slice sampling could be used for separate updates of each parameter, as noted by the authors of the original ESS paper, such an approach would "suffer from the same slow convergence problems as Gibbs sampling." The joint update performed by HSS addresses this concern, but does not suffer from it as the review claims.

- "... I don't think relying on the analytical solution of the Hamiltonian system is a promising direction." We fully agree that relying on the analytical solution of the posterior Hamiltonian system is not a promising direction, but again we strenuously state that HSS is by no means an effort in that direction (see above).

- "Is the comparison between HSS and ESS really necessary? The current results only downgrades the merit of HSS." Yes, this comparison is necessary, for two reasons: 1) as a sanity check that HSS is sensible for a simple model; 2) as stated in the paper, "the ... model allows us to test the performance of HSS against that of ESS on a model for which ESS is ideally suited, forming a highly conservative baseline."

- "The comparison between HMC and HSS is unfair. As mentioned before, HMC can make use of analytical solutions as well." No, this comparison is fair and appropriate.  HMC can not make use of *prior* trajectories, which is what HSS uses (or transformations thereof); the posterior in Section 3.2 is not analytical such that it could be exploited by HMC.

- "Comparisons on real datasets might be more exciting." Our experiments comparing HSS with HMC were performed using real data (the galaxy data set). We are excited by the prospect of further applications on real datasets.