  We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$
with respect to some probability measure. We replace random sampling with
quasi-Monte Carlo methods, where the underlying point set is derived from
deterministic constructions that aim to fill the space more evenly than random
points. Such quasi-Monte Carlo point sets are ordinarily designed for the
uniform measure, and the theory only works for product measures when a
coordinate-wise transformation is applied. Going beyond this setting, we first
consider the case where the target density is a mixture distribution where each
term in the mixture comes from a product distribution. Next we consider target
densities which can be approximated with such mixture distributions. We require
the approximation to be a sum of coordinate-wise products and the approximation
to be positive everywhere (so that they can be re-scaled to probability density
functions). We use tensor product hat function approximations for this purpose
here, since a hat function approximation of a positive function is itself
positive.
  We also study more complex algorithms, where we first approximate the target
density with a general Gaussian mixture distribution and approximate the
mixtures with an adaptive hat function approximation on rotated intervals. The
Gaussian mixture approximation allows us to locate the essential parts of the
target density, whereas the adaptive hat function approximation allows us to
approximate the finer structure of the target density. We prove convergence
rates for each of the integration techniques based on quasi-Monte Carlo
sampling for integrands with bounded partial mixed derivatives. The employed
algorithms are based on digital $(t,s)$-sequences over the finite field
$\mathbb{F}_2$ and an inversion method. Numerical examples illustrate the
performance of the algorithms for some target densities and integrands.

Numerical Analysis;Numerical Analysis;Computation