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I am a novice HMC user. I am reading Neal's chapter in the Handbook of MCMC. I think I can present the HMC algorithm as :

  1. Sample a new momentum
  2. Propose a new momentum and a new position using a reversible and volume preserving integrator such as the Leapfrog Integrator
  3. Accept or reject the proposed momentum and position using a Metropolis step

If I understood well, the critical point in the second step of the HMC algorithm is that the proposal is volume preserving and reversible, but I am free to use another position energy function than the negated log-likelihood of the target. I am aware of the fact that this would be very inefficient because the acceptance rate of HMC depends of the use of Hamiltonian dynamics to visit high-density regions. Is my understanding correct or did I miss something ?

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  • $\begingroup$ Hi: I used to understand this material ( to some extent ) but I can't remember it now. There is an Rstan list ( not stackexchange. google for Rstan ) made up of people who wrote the Rstan language and some of those people over there would probably answer your question. Michael Betancourt is one person who comes to mind. He's very generous and kind. Radford Neal ( not on Rstan ) also might answer if you ask him directly. $\endgroup$ – mlofton Apr 30 '20 at 11:51
  • $\begingroup$ Thanks, I asked on the Stan forum ! $\endgroup$ – SebCoube Apr 30 '20 at 14:55
  • $\begingroup$ Good. I bet that you'll get a good answer from one of those people. My second bet is that it will be Michael Betancourt !!!! $\endgroup$ – mlofton May 1 '20 at 13:21
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If I understood well, the critical point in the second step of the HMC algorithm is that the proposal is volume preserving and reversible, but I am free to use another position energy function than the negated log-likelihood of the target. I am aware of the fact that this would be very inefficient because the acceptance rate of HMC depends of the use of Hamiltonian dynamics to visit high-density regions. Is my understanding correct or did I miss something ?

You are correct that you could use this approach and could still use it to generate a Markov chain with the correct invariant measure. This will usually lead to a worse Markov chain, but may be cheaper if the `wrong Hamiltonian' is faster to compute. An example would be this paper, in which the author uses a cheap surrogate to the desired probability to generate approximate HMC trajectories quickly.

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