General and Efficient Bayesian Computation through Hamiltonian Monte Carlo Extensions

Hamiltonian Monte Carlo (HMC) is a state-of-the-art sampling algorithm for Bayesian computation. Popular probabilistic programming languages Stan and PyMC rely on HMC’s generality and efficiency to provide automatic Bayesian inference platforms for practitioners. Despite its wide-spread use and numerous success stories, HMC has several well known pitfalls. This thesis presents extensions of HMC that overcome its two most prominent weaknesses: inability to handle discrete parameters and slow mixing on multi-modal target distributions.

Discontinuous HMC (DHMC) presented in Chapter 2 extends HMC to discontinuous target distributions – and hence to discrete parameter distributions through embedding them into continuous spaces — using an idea of event-driven Monte Carlo from the computational physics literature. DHMC is guaranteed to outperform a Metropolis-within-Gibbs algorithm since, as it turns out, the two algorithms coincide under a specific (and sub-optimal) implementation of DHMC. The theoretical justification of DHMC extends an existing theory of non-smooth Hamiltonian mechanics and of measure-valued differential inclusions.

Geometrically tempered HMC (GTHMC) presented in Chapter 3 improves HMC’s performance on multi-modal target distributions. The efficiency improvement is achieved through differential geometric techniques, relating a target distribution to

another distribution with less severe multi-modality. We establish a geometric theory behind Riemannian manifold HMC to motivate our geometric tempering methods. We then develop an explicit variable stepsize reversible integrator for simulating

Hamiltonian dynamics to overcome a stability issue of the usual Stormer-Verlet integrator. The integrator is of independent interest, being the first of its kind designed specifically for HMC variants.

In addition to the two extensions described above, Chapter 4 describes a variable trajectory length algorithm that generalizes the acceptance and rejection procedure of HMC — and in fact of any reversible dynamics based samplers — to allow for more flexible choices of trajectory lengths. The algorithm in particular enables an effective application of a variable stepsize integrator to HMC extensions, including GTHMC. The algorithm is widely applicable and provides a recipe for constructing valid dynamics based samplers beyond the known HMC variants. Chapter 5 concludes the thesis with a simple and practical algorithm to improve computational efficiencies of HMC and related algorithms over their traditional implementations.