Monte Carlo methods represent a cornerstone of computer science. They allow to sample high dimensional distribution functions in an efficient way. In this paper we consider the extension of Automatic Differentiation (AD) techniques to Monte Carlo process, addressing the problem of obtaining derivatives (and in general, the Taylor series) of expectation values. Borrowing ideas from the lattice field theory community, we examine two approaches. One is based on reweighting while the other represents an extension of the Hamiltonian approach typically used by the Hybrid Monte Carlo (HMC) and similar algorithms. We show that the Hamiltonian approach can be understood as a change of variables of the reweighting approach, resulting in much reduced variances of the coefficients of the Taylor series. This work opens the door to find other variance reduction techniques for derivatives of expectation values.

翻译：蒙特卡洛方法是计算机科学的基石之一，能够高效地对高维分布函数进行采样。本文考虑将自动微分技术扩展到蒙特卡洛过程，旨在解决期望值导数（以及更一般的泰勒级数）的求解问题。借鉴格点场论领域的思想，我们研究了两种方法：一种基于重加权，另一种则是混合蒙特卡洛（HMC）及类似算法中常用哈密顿方法的推广。我们证明，哈密顿方法可理解为重加权方法中的变量变换，从而显著降低泰勒级数系数的方差。这项工作为寻找期望值导数的其他方差缩减技术开辟了途径。