  We propose a new Markov Chain Monte Carlo (MCMC) method for constrained
target distributions. Our method first maps the $D$-dimensional constrained
domain of parameters to the unit ball ${\bf B}_0^D(1)$. Then, it augments the
resulting parameter space to the $D$-dimensional sphere, ${\bf S}^D$. The
boundary of ${\bf B}_0^D(1)$ corresponds to the equator of ${\bf S}^D$. This
change of domains enables us to implicitly handle the original constraints
because while the sampler moves freely on the sphere, it proposes states that
are within the constraints imposed on the original parameter space. To improve
the computational efficiency of our algorithm, we split the Lagrangian dynamics
into several parts such that a part of the dynamics can be handled analytically
by finding the geodesic flow on the sphere. We apply our method to several
examples including truncated Gaussian, Bayesian Lasso, Bayesian bridge
regression, and a copula model for identifying synchrony among multiple
neurons. Our results show that the proposed method can provide a natural and
efficient framework for handling several types of constraints on target
distributions.
