This paper presents a new minibatch MH algorithm. All the reviewers participated to the discussion after the rebuttal was made available and were unanimously positive about it. * Strengths - The algorithm preserves the correct invariant distribution and shows that algorithms which do not preserve it can go seriously wrong. - The authors provide an interesting theoretical analysis of the algorithm. * Weakenesses - The paper would benefit from a more intuitive introduction of Tuna-MH - A limitation of previous subsampling MH type algorithms is that they appear to be only beneficial in scenarios where the posterior concentrates and is approximately Gaussian as emphasized in Bardenet et al. 2017; Cornish et al., 2019. It should be clarified in the paper whether this is also the case for the method presented here. - The PDMP methods are not only applicable to logistic regression as claimed by the authors.