I do not have many comments, as the paper is technically strong, and the writing is extremely clear. The authors connect very well to previous work. The experimental results are useful to have as well. Thank you for your rebuttal. Overall, even though the paper is well-written, the use of the CLT makes some of the proofs a bit unsatisfactory and hard to check: checking requires a bit more intuition than I would want in a formal context. In the end the paper is making a number of approximations, but since they appear to be useful in practice, I think it is fine as long as the authors emphasize where approximations are made and how their theoretical results could be strengthened,

This work provides a detailed Renyi DP analysis of a modified MCMC acceptance test, and empirically demonstrates its efficacy. Originality: the RDP analysis and modified acceptance test is a novel contribution. Quality: the work is a complete piece on exploring this MCMC method, with a detailed analysis and experiments. Clarity: the work is fairly clearly written, but it can be easy to lose track of exactly what parameters remain as choices to be tuned in a list of various corrective factors and approximations. Significance: the work gives an MCMC method with privacy without convergence, which permits privacy guarantees to be given over a multitude of problems without doubts or guess work about when to stop the chain.

The paper is proposing a DP MCMC algorithm for posterior sampling, adopting the Barker acceptance decomposition and sub-sampling, and by that improving the DP guarantees and the computational requirements. The paper has a rigorous language and it is nicely written. What needs to be more clear from the beginning is the contribution over Seita et al. It takes to page 4 to figure out what this paper is contributing, which is all the results on differential privacy under decomposition and sub-sampling. In addition, the paper needs to clarify what is the aim of having privacy and against which third party?