Vector-Valued Control Variates

Control variates are post-processing tools for Monte Carlo estimators which can lead to significant variance reduction. This approach usually requires a large number of samples, which can be prohibitive for applications where sampling from a posterior 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 integrals jointly. This allows the transfer of information across integration tasks, and hence reduces the overall requirement 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 and model evidence computation through thermodynamic integration.