  Amongst Markov chain Monte Carlo algorithms, Hamiltonian Monte Carlo (HMC) is
often the algorithm of choice for complex, high-dimensional target
distributions; however, its efficiency is notoriously sensitive to the choice
of the integration-time tuning parameter, $T$. When integrating both forward
and backward in time using the same leapfrog integration step as HMC, the set
of local maxima in the potential along a path, or apogees, is the same whatever
point (position and momentum) along the path is chosen to initialise the
integration. We present the Apogee to Apogee Path Sampler (AAPS), which
utilises this invariance to create a simple yet generic methodology for
constructing a path, proposing a point from it and accepting or rejecting that
proposal so as to target the intended distribution. We demonstrate empirically
that AAPS has a similar efficiency to HMC but is much more robust to the
setting of its equivalent tuning parameter, the number of apogees that the path
crosses.
