Vector-Valued Control Variates
Vector-Valued Control Variates
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发表时间：
2021-09
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通讯作者：
Z. Sun;A. Barp;F. Briol
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作者：
Z. Sun;A. Barp;F. Briol
控制变量是蒙特卡罗估计量的方差缩减工具。它们可以提供显著的方差减小，但通常需要大量的样本，当采样或评估被积函数在计算上昂贵时，这可能是禁止的。此外，在许多情况下，我们需要同时或顺序计算多个相关积分，这可能会进一步加剧计算成本。在本文中，我们提出了向量值控制变量，控制变量的扩展，可用于减少多个Monte Carlo估计的方差联合。这允许跨集成任务传输信息，从而减少了对大量样本的需求。我们专注于控制变量的基础上内核插值和我们的新的建设是通过广义Stein身份和发展的新矩阵值Stein再生内核。我们展示了我们的方法，包括多保真度建模，贝叶斯推理的动力系统，并通过热力学集成模型证据计算的一系列问题。
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.