  Numerical Generalized Randomized Hamiltonian Monte Carlo is introduced, as a
robust, easy to use and computationally fast alternative to conventional Markov
chain Monte Carlo methods for continuous target distributions. A wide class of
piecewise deterministic Markov processes generalizing Randomized HMC (Bou-Rabee
and Sanz-Serna, 2017) by allowing for state-dependent event rates is defined.
Under very mild restrictions, such processes will have the desired target
distribution as an invariant distribution. Secondly, the numerical
implementation of such processes, based on adaptive numerical integration of
second order ordinary differential equations (ODEs) is considered. The
numerical implementation yields an approximate, yet highly robust algorithm
that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of
the complete Hamiltonian trajectories (hence the title). The proposed algorithm
may yield large speedups and improvements in stability relative to relevant
benchmarks, while incurring numerical biases that are negligible relative to
the overall Monte Carlo errors. Granted access to a high-quality ODE code, the
proposed methodology is both easy to implement and use, even for highly
challenging and high-dimensional target distributions.
