Spherical Hamiltonian Monte Carlo for Constrained Target Distributions

Abstract

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 ${\mathbf{B}}_0^D(1)$. Then, it augments the resulting parameter space to the $D$-dimensional sphere, ${\mathbf{S}}^D$. The boundary of ${\mathbf{B}}_0^D(1)$ corresponds to the equator of ${\mathbf{S}}^D$. 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.

References in corpus (5)

• MCMC using Hamiltonian dynamics
• Geodesic Monte Carlo on Embedded Manifolds
• Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets
• Lagrangian Dynamical Monte Carlo
• A Semiparametric Bayesian Model for Detecting Synchrony Among Multiple Neurons

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• Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices
• Recycling intermediate steps to improve Hamiltonian Monte Carlo
• Posterior Integration on a Riemannian Manifold
• Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks
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• Efficient Sampling from Feasible Sets of SDPs and Volume Approximation
• Spherical Sliced-Wasserstein
• An Algorithm for Distributed Bayesian Inference in Generalized Linear Models
• An Efficient Scheme for Sampling in Constrained Domains