Bayesian additive adaptive basis tensor product models for modeling high dimensional surfaces: an application to high‐throughput toxicity testing

Many modern datasets are sampled with error from complex high‐dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective and well suited for characterizing a surface in two or three dimensions, but they may suffer from difficulties when representing higher dimensional surfaces. Motivated by high throughput toxicity testing where observed dose‐response curves are cross sections of a surface defined by a chemical's structural properties, a model is developed to characterize this surface to predict untested chemicals’ dose‐responses. This manuscript proposes a novel approach that models the multidimensional surface as a sum of learned basis functions formed as the tensor product of lower dimensional functions, which are themselves representable by a basis expansion learned from the data. The model is described and a Gibbs sampling algorithm is proposed. The approach is investigated in a simulation study and through data taken from the US EPA's ToxCast high throughput toxicity testing platform.