Why is it important for HMC to keep the total Energy constant during the sampling using the leapfrog integration?
What is the consequence of a big energy change at the end of the integration?
Proposals that conserve energy are always accepted.
If energy isn’t conserved, proposals can be accepted or rejected in a Metropolis-Hastings step. The fact that proposals can be rejected means that you explore the posterior more slowly, though you can still obtain correct results.
Thanks for the answer. Could you explain why we need to conserve the total energy during the numerical integration? What is the consequence of the deviation from the starting point energy?
I am not clear on the reason why we need to keep the energy constant here?
I roughly understand HMC provides us a way to sample more efficiently to get more uncorrelated samples in a markov chain. My gut feeling is that the numerical leapfrog integration in many steps will cause randomness during the sampling so it seems a bit deviation of energy will not cause much issue.
I suggest you read
We don’t need to conserve energy; but it’s more efficient if we do. The bigger the drops in energy, the more chance that proposals are rejected and the less efficient sampling becomes.
I recommend for reading also
- An introduction for applied users with good visualizations: Monnahan, Thorson, and Branch (2016). Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo. https://dx.doi.org/10.1111/2041-210X.12681
- A technical review of why HMC works: Neal (2012). MCMC using Hamiltonian dynamics. [1206.1901] MCMC using Hamiltonian dynamics