  Bayesian quadrature (BQ) is a sample-efficient probabilistic numerical method
to solve integrals of expensive-to-evaluate black-box functions, yet so
far,active BQ learning schemes focus merely on the integrand itself as
information source, and do not allow for information transfer from cheaper,
related functions. Here, we set the scene for active learning in BQ when
multiple related information sources of variable cost (in input and source) are
accessible. This setting arises for example when evaluating the integrand
requires a complex simulation to be run that can be approximated by simulating
at lower levels of sophistication and at lesser expense. We construct
meaningful cost-sensitive multi-source acquisition rates as an extension to
common utility functions from vanilla BQ (VBQ),and discuss pitfalls that arise
from blindly generalizing. Furthermore, we show that the VBQ acquisition policy
is a corner-case of all considered cost-sensitive acquisition schemes, which
collapse onto one single de-generate policy in the case of one source and
constant cost. In proof-of-concept experiments we scrutinize the behavior of
our generalized acquisition functions. On an epidemiological model, we
demonstrate that active multi-source BQ (AMS-BQ) allocates budget more
efficiently than VBQ for learning the integral to a good accuracy.
