Riemannian manifold Hamiltonian Monte Carlo (RMHMC) is a powerful method of Bayesian inference that exploits underlying geometric information of the posterior distribution in order to efficiently traverse the parameter space. However, the form of the Hamiltonian necessitates complicated numerical integrators, such as the generalized leapfrog method, that preserve the detailed balance condition. The distinguishing feature of these numerical integrators is that they involve solutions to implicitly defined equations. Lagrangian Monte Carlo (LMC) proposes to eliminate the fixed point iterations by transitioning from the Hamiltonian formalism to Lagrangian dynamics, wherein a fully explicit integrator is available. This work makes several contributions regarding the numerical integrator used in LMC. First, it has been claimed in the literature that the integrator is only first-order accurate for the Lagrangian equations of motion; to the contrary, we show that the LMC integrator enjoys second order accuracy. Second, the current conception of LMC requires four determinant computations in every step in order to maintain detailed balance; we propose a simple modification to the integration procedure in LMC in order to reduce the number of determinant computations from four to two while still retaining a fully explicit numerical integration scheme. Third, we demonstrate that the LMC integrator enjoys a certain robustness to human error that is not shared with the generalized leapfrog integrator, which can invalidate detailed balance in the latter case. We discuss these contributions within the context of several benchmark Bayesian inference tasks.
翻译:拉格兰吉·蒙特卡洛(RMHC)是一个强有力的巴伊西亚推论方法,它利用了后方分布的几何信息基础,以有效穿越参数空间。然而,汉密尔顿式的形态需要复杂的数字集成器,例如通用的跳蛙法,以保持详细的平衡条件。这些数字集成器的显著特征是它们涉及隐含定义方程式的解决办法。拉格兰吉安·蒙特卡洛(LMC)建议消除固定点交接点,从汉密尔顿正式主义过渡到拉格兰格的动态,其中有一个完全明确的集成器。这项工作对LMC所使用的数字集成器作出了若干贡献。首先,文献中声称,这些集成器只是对拉格兰加方方方方方方方程式的第一阶准确度;相反,我们显示LMC调成体具有第二级的精确度。第二,目前对LMC概念的概念要求用四个决定因素进行计算,以便每一步保持详细的平衡;我们建议对LMC中使用的数字集成两个数字集成法进行彻底的计算,而我们又建议将这些数字集成一个数字集成的精度的精度进行精确的分。