I initially posted the following question on Stackexchange, and a member advised me to come and ask it here.

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 :

- Sample a new momentum
- Propose a new momentum and a new position using a reversible and volume preserving integrator such as the Leapfrog Integrator
- 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. However, there might be an interest if someone was using a cheap approximation to the density, or to use a heated density to do some tempering (the option is not mentioned in the manual though). Is my understanding correct or did I miss something ?