  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, and on a Bayesian survey design problem.
