In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of HMC which aims at sampling from a Gibbs distribution $\pi$ on a manifold $\mathsf{M}$, endowed with a Hessian metric $\mathfrak{g}$ derived from a self-concordant barrier. Like Riemannian Manifold HMC, our method relies on Hamiltonian dynamics which comprise $\mathfrak{g}$. It incorporates the constraints defining $\mathsf{M}$ and is therefore able to exploit its underlying geometry. We first introduce c-BHMC (continuous BHMC), for which we assume that the Hamiltonian dynamics can be integrated exactly, and show that it generates a Markov chain for which $\pi$ is invariant. Secondly, we design n-BHMC (numerical BHMC), a Metropolis-Hastings algorithm which combines an acceptance filter including a "reverse integration check" and numerical integrators of the Hamiltonian dynamics. Our main results establish that n-BHMC generates a reversible Markov chain with respect to $\pi$. This is in contrast to existing algorithms which extend the HMC method to Riemannian manifolds, as they do not deal with asymptotic bias. Our conclusions are supported by numerical experiments where we consider target distributions defined on polytopes.
翻译:在本文中,我们提出“屏障汉密尔顿蒙特卡洛 ” (BHMC),这是HMC的版本,目的是从Gibbs分发的$\pi$上采样一个元$mathsf{MM}$,由自相调合的屏障制成。像Riemannian Maniflex HMC一样,我们的方法依赖汉密尔顿的动态,由$\mathfrak{g}美元构成。它包含了对美元定义的制约,因此能够对其基本几何进行利用。我们首先引入了C-BHMC(连续BHMC),为此我们假定汉密尔顿的动态可以完全整合,并显示它会产生一个马克夫链,而美元则是无变动的。第二,我们设计了n-BHMCC(数字BHMC),一个Metrobolpolis-Has算法,它结合了一种接受过滤器,包括“反向融合检查”和汉密尔顿动态的数字拼。我们的主要结果是n-BHMMC(我们目前定义的比卡式)比卡路路段的比比。