  Pre-integration is an extension of conditional Monte Carlo to quasi-Monte
Carlo and randomized quasi-Monte Carlo. It can reduce but not increase the
variance in Monte Carlo. For quasi-Monte Carlo it can bring about improved
regularity of the integrand with potentially greatly improved accuracy.
Pre-integration is ordinarily done by integrating out one of $d$ input
variables to a function. In the common case of a Gaussian integral one can also
pre-integrate over any linear combination of variables. We propose to do that
and we choose the first eigenvector in an active subspace decomposition to be
the pre-integrated linear combination. We find in numerical examples that this
active subspace pre-integration strategy is competitive with pre-integrating
the first variable in the principal components construction on the Asian option
where principal components are known to be very effective. It outperforms other
pre-integration methods on some basket options where there is no well
established default. We show theoretically that, just as in Monte Carlo,
pre-integration can reduce but not increase the variance when one uses
scrambled net integration. We show that the lead eigenvector in an active
subspace decomposition is closely related to the vector that maximizes a less
computationally tractable criterion using a Sobol' index to find the most
important linear combination of Gaussian variables. They optimize similar
expectations involving the gradient. We show that the Sobol' index criterion
for the leading eigenvector is invariant to the way that one chooses the
remaining $d-1$ eigenvectors with which to sample the Gaussian vector.
