Bayesian quadrature (BQ) is a method for solving numerical integration problems in a Bayesian manner, which allows users to quantify their uncertainty about the solution. The standard approach to BQ is based on a Gaussian process (GP) approximation of the integrand. As a result, BQ is inherently limited to cases where GP approximations can be done in an efficient manner, thus often prohibiting very high-dimensional or non-smooth target functions. This paper proposes to tackle this issue with a new Bayesian numerical integration algorithm based on Bayesian Additive Regression Trees (BART) priors, which we call BART-Int. BART priors are easy to tune and well-suited for discontinuous functions. We demonstrate that they also lend themselves naturally to a sequential design setting and that explicit convergence rates can be obtained in a variety of settings. The advantages and disadvantages of this new methodology are highlighted on a set of benchmark tests including the Genz functions, on a rare-event simulation problem and on a Bayesian survey design problem.
Harrison Zhu (Imperial College London)
I am a PhD student at the Imperial College London in Modern Statistics and Statistical Machine Learning. Interested in Bayesian statistics, theoretical statistics and causal inference for earth sciences, survey design, experimental design and spatial statistics.
Xing Liu (Imperial College London)
Ruya Kang (Brown University)
Zhichao Shen (University of Oxford)
Seth Flaxman (Imperial College London)
Francois-Xavier Briol (University of Cambridge)
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