This paper addresses the question of making inferences regarding Boolean functions under conditions of (i) noise, or stochastic variation in observed data, and (ii) sparsity, by which we mean that the number of inputs or predictors far exceeds the arity of the underlying Boolean function. We put forward a simple probability model for such observations, and discuss model selection, parameter estimation and prediction. We present results on synthetic data and on a proteomic dataset from a study in cancer systems biology.