Sparse combinatorial inference with an application in cancer biology

Motivation: 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. Results: We put forward a statistical model for sparse, noisy Boolean functions and methods for inference under the model. We focus on the case in which the form of the underlying Boolean function, as well as the number and identity of its inputs are all unknown. We present results on synthetic data and on a study of signalling proteins in cancer biology.