  We present an improved Bayesian framework for performing inference of affine
transformations of constrained functions. We focus on quadrature with
nonnegative functions, a common task in Bayesian inference. We consider
constraints on the range of the function of interest, such as nonnegativity or
boundedness. Although our framework is general, we derive explicit
approximation schemes for these constraints, and argue for the use of a log
transformation for functions with high dynamic range such as likelihood
surfaces. We propose a novel method for optimizing hyperparameters in this
framework: we optimize the marginal likelihood in the original space, as
opposed to in the transformed space. The result is a model that better explains
the actual data. Experiments on synthetic and real-world data demonstrate our
framework achieves superior estimates using less wall-clock time than existing
Bayesian quadrature procedures.
