Reversibility is a *“second-order”* property for MCMC algorithms. E.g., the Gibbs sampler is not reversible but does work without that property… In addition, the standard Metropolis-Hastings scheme is reversible by construction (courtesy of the *detailed balance* property). My point in that post (and in the incoming discussion) is that, similar to Langevin, the continuous time “analogy” may be unnecessary costly in trying to carry this analogy in discrete (algorithmic) time… We are not working in continuous time, the invariance/stability properties of the continuous time process are not carried on to the discretised version of the process, we therefore should not care about reproducing exactly the continuous time process in discrete time. For instance, when considering the Langevin diffusion, the corresponding Langevin algorithm could use another scale for the gradient than the one used for the noise, i.e.

rather than the Euler discretisation

A few experiments at the time of the first edition of MCSM (Chapter 6, Section 6.5) showed that a different scale could lead to improvements.

]]>Besides, wouldn’t this integrator still have to be reversible ? Or would the Metropolis-Hastings rejection correct this too ? [the potential lack of “volume-preservation” (i.e. Jacobian \neq 1) is not a problem, as we can introduce the Jacobian in the MH ratio [assuming it can be computed !], similar to what’s done in Reversible Jump if I remember well].

The original version of the paper ( http://arxiv.org/pdf/0907.1100v1 ) had a (now removed) additional author and a (now removed) closed-form integrator. Although this integrator has been pulled because it was flawed (scientific grapevine hearsay, and I don’t know what the flaw was — nor did I check it, I openly confess), we can check the simulation results it gave, on the same examples, for an indication of the performance drop — except if the flaw was a Jacobian issue, which of course would have flawed the MH ratio.

I’ll call Mark’s attention on this thread — but he might want to save his comments for the discussion answers.

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