Control variates are variance reduction tools for Monte Carlo estimators. They can provide significant variance reduction, but usually require a large number of samples, which can be prohibitive when sampling or evaluating the integrand is computationally expensive. Furthermore, there are many scenarios where we need to compute multiple related integrals simultaneously or sequentially, which can further exacerbate computational costs. In this paper, we propose vector-valued control variates, an extension of control variates which can be used to reduce the variance of multiple Monte Carlo estimators jointly. This allows for the transfer of information across integration tasks, and hence reduces the need for a large number of samples. We focus on control variates based on kernel interpolants and our novel construction is obtained through a generalised Stein identity and the development of novel matrix-valued Stein reproducing kernels. We demonstrate our methodology on a range of problems including multifidelity modelling, Bayesian inference for dynamical systems, and model evidence computation through thermodynamic integration.

翻译：控制变量法是蒙特卡洛估计的方差缩减工具。该方法可显著降低方差，但通常需要大量样本，当采样或评估被积函数计算成本较高时，这种需求可能难以实现。此外，许多场景需要同时或依次计算多个相关积分，这会进一步加剧计算负担。本文提出向量值控制变量法，作为传统控制变量法的扩展，可联合减少多个蒙特卡洛估计的方差。该方法能够跨积分任务传递信息，从而降低对大量样本的需求。我们重点研究基于核插值的控制变量，通过广义Stein恒等式与新型矩阵值Stein再生核的构建实现这一创新方法。我们在多保真度建模、动力系统贝叶斯推断以及热力学积分模型证据计算等系列问题中验证了所提方法的有效性。