  We propose a Bayesian regression method that accounts for multi-way
interactions of arbitrary orders among the predictor variables. Our model makes
use of a factorization mechanism for representing the regression coefficients
of interactions among the predictors, while the interaction selection is guided
by a prior distribution on random hypergraphs, a construction which generalizes
the Finite Feature Model. We present a posterior inference algorithm based on
Gibbs sampling, and establish posterior consistency of our regression model.
Our method is evaluated with extensive experiments on simulated data and
demonstrated to be able to identify meaningful interactions in applications in
genetics and retail demand forecasting.
