Combinatorial effects, in which several variables jointly influence an output or response, play an important role in biological systems. In many settings, Boolean functions provide a natural way to describe such influences. However, biochemical data using which we may wish to characterize such influences are usually subject to much variability. Furthermore, in high-throughput biological settings Boolean relationships of interest are very often sparse, in the sense of being embedded in an overall dataset of higher dimensionality. This motivates a need for statistical methods capable of making inferences regarding Boolean functions under conditions of noise and sparsity.