In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the HMC algorithm which aims at sampling from a Gibbs distribution $\pi$ on a manifold $\mathrm{M}$, endowed with a Hessian metric $\mathfrak{g}$ derived from a self-concordant barrier. Our method relies on Hamiltonian dynamics which comprises $\mathfrak{g}$. Therefore, it incorporates the constraints defining $\mathrm{M}$ and is able to exploit its underlying geometry. However, the corresponding Hamiltonian dynamics is defined via non separable Ordinary Differential Equations (ODEs) in contrast to the Euclidean case. It implies unavoidable bias in existing generalization of HMC to Riemannian manifolds. In this paper, we propose a new filter step, called "involution checking step", to address this problem. This step is implemented in two versions of BHMC, coined continuous BHMC (c-BHMC) and numerical BHMC (n-BHMC) respectively. Our main results establish that these two new algorithms generate reversible Markov chains with respect to $\pi$ and do not suffer from any bias in comparison to previous implementations. Our conclusions are supported by numerical experiments where we consider target distributions defined on polytopes.

翻译：本文提出屏障哈密顿蒙特卡洛（BHMC）算法，该算法是HMC算法的一个变体，旨在从赋予由自和谐壁垒导出的海森度量$\mathfrak{g}$的流形$\mathrm{M}$上的吉布斯分布$\pi$中进行采样。我们的方法依赖于包含$\mathfrak{g}$的哈密顿动力学。因此，它能够融入定义$\mathrm{M}$的约束条件，并利用其底层几何结构。然而，与欧几里得情况不同，相应的哈密顿动力学是通过非可分常微分方程（ODEs）定义的。这意味着现有的将HMC推广到黎曼流形的方法存在不可避免的偏差。在本文中，我们提出了一种新的滤波步骤，称为“对合检查步骤”，以解决这一问题。该步骤在BHMC的两个版本中实现，分别称为连续BHMC（c-BHMC）和数值BHMC（n-BHMC）。我们的主要结果表明，这两个新算法生成的关于$\pi$的马尔可夫链是可逆的，并且与之前的实现相比不存在任何偏差。我们的结论得到了数值实验的支持，其中我们考虑了定义在多胞体上的目标分布。