用Velocity Verlet积分器改进HMC抽样方法

Improving the HMC sampling method by Velocity Verlet integrator

摘要:

Hamilton Monte Carlo (HMC) 方法是一种常用的快速抽样方法. 在对哈密尔顿方程进行抽样时，HMC方法使用Leapfrog积分器，这可能造成方程的位置和动量的迭代值在时间上不同步，产生的误差会降低抽样效率及抽样结果的稳定性. 为此，本文提出了IHMC（Improved HMC）方法，该方法用Velocity Verlet积分器代替Leapfrog积分器，在每次迭代时都计算两变量在同一时刻的值. 为验证方法的效果，本文进行了两个数值实验，一个是将方法应用于非对称随机波动率模型（RASV模型）的参数估计，另一个是将方法应用于方差伽马分布的抽样. 结果显示，IHMC方法比HMC方法的效率更高，结果更稳定.

Abstract:

Hamilton Monte Carlo (HMC) method is a fast sampling algorithm. To sampling a Hamiltonian equation, the HMC method uses the Leapfrog integrator. As a result, the unsynchronized values of position and momentum variables of the equaiton are possibly generated during the iterative process and the resulting estimation errors can seriously damage the sampling efficiency and stability of the sampling result. To solve this problem, here we propose an improved HMC (IHMC) method by replacing the Leapfrog integrator in HMC with the Velocity Verlet integrator, The IHMC method calculates the value of both variables simutaneously in each iteration. Two numerical examples are implemented to check the performance of IHMC method, one is the problem of estiimating the parameters of realized stochastic volatility (RASV) model with asymmetric effects, the other is the problem of sampling the variance Gamma distribution. It is shown that the IHMC method has higher sampling efficiency and more stable sampling result compared with the HMC method.