Control variates with a dimension reduced Bayesian Monte Carlo sampler

Evaluating the expectations of random functions is an important task in many fields of science and engineering. In practice, such an expectation is often evaluated with the Monte Carlo methods which rely on approximating the sought expectation with a sample average. It is well known that the Monte Carlo methods typically suffer from a slow convergence, which makes it especially undesirable for problems where generating samples requires expensive computer simulations. An alternative to the standard Monte Carlo methods is the so-called Bayesian Monte Carlo algorithm, which formulates the expectation estimation as a Bayesian inference problem. As has been demonstrated in literature, the Bayesian Monte Carlo method is often more efficient than the standard Monte Carlo in a large range of problems. However, a major limitation of Bayesian Monte Carlo is that it models the integrand function as a Gaussian process, and as a result, it can not handle problems of high dimension. In this work, we propose a method to address this issue, and specifically we incorporate the Bayesian Monte Carlo framework with the likelihood-based dimension reduction technique, which allows us to evaluate the expectation of functions with very high dimension. In addition, we also provide a control variate scheme to further improve the performance in case the dimension reduction or the BMC estimation is not accurate. We then apply the proposed method to compute the Bayesian evidence in large-scale inference problems.