We show how the Hamiltonian Monte Carlo algorithm can sometimes be speeded up by "splitting" the Hamiltonian in a way that allows much of the movement around the state space to be done at low computational cost. One context where this is possible is when the log density of the distribution of interest (the potential energy function) can be written as the log of a Gaussian density, which is a quadratic function, plus a slowly varying function. Hamiltonian dynamics for quadratic energy functions can be analytically solved. With the splitting technique, only the slowly-varying part of the energy needs to be handled numerically, and this can be done with a larger stepsize (and hence fewer steps) than would be necessary with a direct simulation of the dynamics. Another context where splitting helps is when the most important terms of the potential energy function and its gradient can be evaluated quickly, with only a slowly-varying part requiring costly computations. With splitting, the quick portion can be handled with a small stepsize, while the costly portion uses a larger stepsize. We show that both of these splitting approaches can reduce the computational cost of sampling from the posterior distribution for a logistic regression model, using either a Gaussian approximation centered on the posterior mode, or a Hamiltonian split into a term that depends on only a small number of critical cases, and another term that involves the larger number of cases whose influence on the posterior distribution is small. Supplemental materials for this paper are available online.
Identifying the Optimal Integration Time in Hamiltonian Monte Carlo
通过利用光滑概率系统的自然几何结构，哈密顿蒙特卡罗产生计算效率高的马尔可夫链蒙特卡罗估计。至少在算法足够好的情况下。在本文中，我将展示如何几何基础的哈密顿蒙特卡罗隐含地确定这些参数的最佳选择，特别是积分时间。然后，我认为这些原则在现有的算法和一个新的实现称为\emph{Exhaustive Hamiltonian Monte Carlo}的实际后果，然后在一些说明性的例子证明这些想法的效用。
Some Contributions to Sequential Monte Carlo Methods for Option Pricing
期权定价是金融工程中的一个重要问题。在许多具有实际意义的场景中，与标的资产相关的金融期权价格简化为计算w.r.t.~扩散过程。一般来说，这些期望值无法通过分析计算，而近似这些数量的一种方法是通过蒙特卡罗方法;蒙特卡罗方法至少从20世纪70年代开始就被用于期权定价。在Del Moral，P.\& Shevchenko，P.V.（2014）"Valuation of barrier options using序贯蒙特卡罗"和Jasra，A. \& Del Moral，P.（2011）"期权定价的序贯蒙特卡罗"，序贯蒙特卡罗（SMC）方法是应用于这种情况的自然工具，可以大大改善标准蒙特卡罗。在本文中，本着类似的精神，Del Moral，P.\& Shevchenko，P.V.（2014）"Valuation of barrier options using序贯蒙特卡罗"和Jasra，A. \& Del Moral，P.（2011）"期权定价的序贯蒙特卡罗"我们表明，通过构建一系列人工目标密度随时间的推移使用SMC方法可以实现显着收益。特别是，我们近似的最佳重要性抽样分布的SMC算法，通过使用一系列的加权函数。这是证明了两个例子，障碍期权和目标应计赎回票据（TARN的）。我们还提供了我们的SMC估计的无偏性的证明。
