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This paper should be regarded asatheoretical contribution that takesacritical stand on the "usual assumptions" on4 which PAC-Bayes inequalities are based (i.e. ResponsetoReviewer222 We will do a better job at discussing the usual formulation (for all posterior distributions) versus the formulation in23 terms ofastochastic kernel. Note that24 each theorem holds for an arbitrary kernel, so in that sense it holds "for all posteriors" as per the usual formulation25 in the literature.