  This paper develops a comprehensive probabilistic setup to compute
approximating functions in active subspaces. Constantine et al. proposed the
active subspace method in (Constantine et al., 2014) to reduce the dimension of
computational problems. It can be seen as an attempt to approximate a
high-dimensional function of interest $f$ by a low-dimensional one. To do this,
a common approach is to integrate $f$ over the inactive, i.e. non-dominant,
directions with a suitable conditional density function. In practice, this can
be done with a finite Monte Carlo sum, making not only the resulting
approximation random in the inactive variable for each fixed input from the
active subspace, but also its expectation, i.e. the integral of the
low-dimensional function weighted with a probability measure on the active
variable. In this regard we develop a fully probabilistic framework extending
results from (Constantine et al., 2014, 2016). The results are supported by a
simple numerical example.
