Bayesian Quadrature for Multiple Related Integrals

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.Comment: Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR 80:5369-5378, 201

A Bayesian Monte Carlo Approach to Global Illumination

Most Monte Carlo rendering algorithms rely on importance sampling to reduce the variance of estimates. Importance sampling is efficient when the proposal sample distribution is well-suited to the form of the integrand but fails otherwise. The main reason is that the sample location information is not exploited. All sample values are given the same importance regardless of their proximity to one another. Two samples falling in a similar location will have equal importance whereas they are likely to contain redundant information. The Bayesian approach we propose in this paper uses both the location and value of the data to infer an integral value based on a prior probabilistic model of the integrand. The Bayesian estimate depends only on the sample values and locations, and not how these samples have been chosen. We show how this theory can be applied to the final gathering problem and present results that clearly demonstrate the benefits of Bayesian Monte Carlo