Automatic cubatures approximate multidimensional integrals to user-specified error tolerances. For high dimensional problems, it makes sense to fix the sampling density but determine the sample size, n, automatically. Bayesian cubature postulates that the integrand is an instance of a stochastic process. Here we assume a Gaussian process parameterized by a constant mean and a covariance function defined by a scale parameter times a parameterized function specifying how the integrand values at two different points in the domain are related. These parameters are estimated from integrand values or are given non-informative priors. The sample size, n, is chosen to make the half-width of the credible interval for the Bayesian posterior mean no greater than the error tolerance.The process just outlined typically requires vectormatrix operations with a computational cost of O(n 3 ). Our innovation is to pair low discrepancy nodes with matching kernels that lower the computational cost to O(n log n). This approach is demonstrated using rank-1 lattice sequences and shift-invariant kernels. Our algorithm is implemented in the Guaranteed Automatic Integration Library (GAIL).

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Probabilistic integration provides a criterion for stopping a simulation when a specified error tolerance is satisfied with high confidence. We comment on some of the modeling assumptions and implementation issues involved in designing an automatic Bayesian cubature.

Function approximation, integration, and optimization are three fundamental mathematical problems. They are especially challenging when the functions involved fluctuate wildly in certain parts of the domain, or if the domain is high dimensional. Ideally, algorithms to solve these problems should possess a rigorous mathematical framework, data-based (probabilistic) error bounds, and advanced sampling strategies for efficiency.

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