  Automatic cubatures approximate integrals to user-specified error tolerances.
For high dimensional problems, it is difficult to adaptively change the
sampling pattern to focus on peaks because peaks can hide more easily in high
dimensional space. But, one can automatically determine the sample size, $n$,
given a reasonable, fixed sampling pattern. This approach is pursued in
Jagadeeswaran and Hickernell, Stat.\ Comput., 29:1214-1229, 2019, where a
Bayesian perspective is used to construct a credible interval for the integral,
and the computation is terminated when the half-width of the interval is no
greater than the required error tolerance. Our earlier work employs integration
lattice sampling, and the computations are expedited by the fast Fourier
transform because the covariance kernels for the Gaussian process prior on the
integrand are chosen to be shift-invariant. In this chapter, we extend our fast
automatic Bayesian cubature to digital net sampling via \emph{digitally}
shift-invariant covariance kernels and fast Walsh transforms.
  Our algorithm is implemented in the MATLAB Guaranteed Automatic Integration
Library (GAIL) and the QMCPy Python library.
