Title: The apogee to apogee path sampler

Abstract: Hamiltonian Monte Carlo (HMC) is often the method of choice when performing inference on complex high-dimensional Bayesian posterior distributions; however, it is notoriously difficult to tune. In particular, slight changes in the choice of integration time can make the difference between an optimally efficient algorithm and a poor one. We define an \emph{apogee} of a Hamiltonian path as a point in the path where the component of momentum in the direction of the gradient of the potential changes from positive to negative; essentially, the “ball” is about to change from rolling up hill to rolling down hill. From this we introduce the Apogee to Apogee Path Sampler, which uses the same leapfrog dynamics as HMC but eschews the integration time parameter in favour of the choice of a number of apogees, and allows for a variety of proposal mechanisms that choose from the whole Hamiltonian path. The resulting algorithm is competitive with optimally tuned HMC, and is more robust to the tuning choice of “number of apogees” than HMC is to choice of integration time. Furthermore, the number of apogees relates directly to intrinsic properties of the posterior, and allows for tuning guidelines from an initial run. Finally, the algorithm requires no self-recursions and for certain useful classes of proposal the storage cost through an iteration is $\mathcal{O}(1)$