# Proposal distribution in Hamiltonian Monte Carlo

I have been reading A Conceptual Introduction to Hamiltonian Monte Carlo by Betancourt (https://arxiv.org/abs/1701.02434), which is a great introduction to HMC, but there is one part that I can't get my head around and that's what the proposal distribution in HMC is. In standard Metropolis Hastings we can use a normal or uniform distribution as proposal distribution, and they cancel because of symmetry. This is also the case in HMC, but I can't see what the actual distribution is. Is it the momentum variable (i.e multivariate normal)? Can anybody explain this or link to sources that explain it? Thanks!

The proposal distribution for the original Hamiltonian Monte Carlo algorithm is just a delta function around the final point in the numerical trajectory with the momentum negated, $$K(z' | z) = \delta \, (z' - R(\Phi_{\epsilon, L}(z))),$$ where $z = (q, p)$ is a point on phase space, $\Phi_{\epsilon, L}(z)$ is the action of the numerical integrator, and $R$ is the negation operator that fits the sign of the momentum, $$R(q, p) = (q, -p).$$
Importantly, the original Hamiltonian Monte Carlo algorithm cannot be interpreted as a Metropolis-Hastings algorithm on the target parameter space and so there is no proposal distribution $K(q'|q)$. The Metropolis-Hastings acceptance procedure has to be done on the extended phase space that includes the auxiliary momenta in addition to the target parameters. The overall procedure of sampling momenta to generate a point in phase space, numerically integrating a trajectory, accepting or rejecting the final point in phase space, and then throwing the momenta away to recover a new parameter value, however, defines a Markov kernel on the target parameter space.