  By leveraging the natural geometry of a smooth probabilistic system,
Hamiltonian Monte Carlo yields computationally efficient Markov Chain Monte
Carlo estimation. At least provided that the algorithm is sufficiently
well-tuned. In this paper I show how the geometric foundations of Hamiltonian
Monte Carlo implicitly identify the optimal choice of these parameters,
especially the integration time. I then consider the practical consequences of
these principles in both existing algorithms and a new implementation called
\emph{Exhaustive Hamiltonian Monte Carlo} before demonstrating the utility of
these ideas in some illustrative examples.
