In neuroscience, attention has been associated operationally with enhanced processing of certain sensory inputs depending on external or internal contexts such as cueing, salience, or mental states. In machine learning, attention usually means a multiplicative mechanism whereby the weights in a weighted summation of an input vector are calculated from the input itself or some other context vector. In both scenarios, attention can be conceptualized as a gating mechanism. In this paper, we argue that three-way interactions serve as a normative way to define a gating mechanism in generative probabilistic graphical models. By going a step beyond pairwise interactions, it empowers much more computational efficiency, like a transistor expands possible digital computations. Models with three-way interactions are also easier to scale up and thus to implement biologically. As an example application, we show that a graphical model with three-way interactions provides a normative explanation for divisive normalization in macaque primary visual cortex, an operation adopted widely throughout the cortex to reduce redundancy, save energy, and improve computation.