When I read the paper by Neal, 'MCMC using Hamiltonian dynamics, https://arxiv.org/pdf/1206.1901.pdf In page 4, final paragraph, it says

'The reversibility of Hamiltonian dynamics is important for showing that MCMC updates that use the dynamics leave the desired distribution invariant, since this is most easily proved by showing reversibility of the Markov chain transitions, which requires reversibility of the dynamics used to propose a state. '

My understanding is that the Metropolis Hasting algorithm has guaranteed the Markov chain to be reversible, then why we still need the property of reversibility of Hamiltonian dynamics?

  • $\begingroup$ I think the quote says it all: reversibility is an easy way to prove convergence, not a property to be sought. It applies to Metropolis-Hastings and it applies to HMC.. $\endgroup$ – Xi'an Mar 2 '17 at 12:17
  • $\begingroup$ Thanks @Xi'an, my understanding now is that the Hamiltonian dynamics can generate deterministic samples which has invariant distribution corresponding to the target distribution, however, the reversibility can be destroyed by the numerical simulation of the evaluation equation (Hamiltonian dynamics), then we still adapt the Metropolis-Hastings to keep the chain reversibility. $\endgroup$ – Fly_back Mar 3 '17 at 1:30

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