  We explore the construction of new symplectic numerical integration schemes
to be used in Hamiltonian Monte Carlo and study their efficiency. Two
integration schemes from Blanes et al. (2014), and a new scheme based on
optimal acceptance probability, are considered as candidates to the commonly
used leapfrog method. All integration schemes are tested within the framework
of the No-U-Turn sampler (NUTS), both for a logistic regression model and a
student $t$-model. The results show that the leapfrog method is inferior to all
the new methods both in terms of asymptotic expected acceptance probability for
a model problem and the and efficient sample size per computing time for the
realistic models.
