| Modern statistical methods relying on Bayesian inference typically involve intractable models that require computationally intensive algorithms, such as Markov Chain Monte Carlo (MCMC), for their implementation. While simple MCMC algorithms (e.g., random walk Metropolis) might be effective at exploring low-dimensional probability distributions, they can be very inefficient for complex, high-dimensional distributions. More specifically, broader application of MCMC is hindered by either slow mixing rate or expensive computational cost. As a result, many existing MCMC algorithms are not efficient or capable enough to handle complex models that are now commonly used in statistics and machine learning. This dissertation focuses on utilizing geometrically motivated methods to improve efficiency of MCMC samplers while lowering the computational cost, with the aim to extend the application of MCMC methods to complex statistical problems involving heavy computation, complicated distribution structure, multimodality, and parameter constraints.;We start by extending the standard Hamiltonian Monte Carlo (HMC) algorithm through splitting the Hamiltonian in a way that allows much of the movement around the state space achieved at low computational cost. For more advanced HMC algorithms defined on Riemannian manifolds, we propose a new method, Lagrangian Monte Carlo, which is capable of exploring complex probability distributions at relatively low computational cost. For multimodal distributions, we have developed a geometrically motivated approach, Wormhole Hamiltonian Monte Carlo, that explores the distribution around the known modes effectively while identifying previously unknown modes in the process. Furthermore, we propose another algorithm, Spherical Hamiltonian Monte Carlo, that combines geometric methods and computational techniques to provide a natural and efficient framework for sampling from constrained distributions. We use a variety of simulations and real data to illustrate the substantial improvement obtained by our proposed methods over alternative solutions. |
