Spherical Hamiltonian Monte Carlo for constrained target distributions

Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit models, many copula models, and Latent Dirichlet Allocation (LDA) models. Bayesian inference involving probability distributions confined to constrained domains could be quite challenging for commonly used sampling algorithms. For such problems, we propose a novel Markov Chain Monte Carlo (MCMC) method that provides a general and computationally efficient framework for handling boundary conditions. Our method first maps the -D-dimensional constrained domain of parameters to the unit ball B^D₀(1), then augments it to a D-dimensional sphere S^D such that the original boundary corresponds to the equator of S^D. This way, our method handles the constraints implicitly by moving freely on the sphere generating proposals that remain within boundaries when mapped back to the original space.