Let me clarify the part of the paper that is unclear: “The paper contains an exact comparison result that non-reversible jump leads to a smaller asymptotic variance than reversible jump, but it is unclear to me whether or not this accounts for the extra computing time resulting from the multiple paths in the proposed algorithms.” Consider that P_NRJ is the Markov kernel simulated by any non-reversible jump (NRJ) algorithms (e.g., the vanilla version without the approaches of Andrieu) and P_RJ that simulated by its reversible counterpart. If P_NRJ is the Markov kernel simulated by the vanilla NRJ, then P_RJ is the Markov kernel simulated by the vanilla RJ. These algorithms have the same computational cost because using a direction/spin does not increase the computation cost. Our result allows to compare these two algorithms in terms of asymptotic variance.

Our result does not allow to compare the vanilla NRJ with the RJ using the approaches of Andrieu, for instance. As you mentioned, we compared such algorithms on a numerical example (in which all algorithms are given the same computational budget).

Best regards,

Philippe

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