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Discovering interaction effects on a response of interest is a fundamentalproblem faced in biology, medicine, economics, and many other scientificdisciplines. In theory, Bayesian methods for discovering pairwise interactionsenjoy many benefits such as coherent uncertainty quantification, the ability toincorporate background knowledge, and desirable shrinkage properties. Inpractice, however, Bayesian methods are often computationally intractable foreven moderate-dimensional problems. Our key insight is that many hierarchicalmodels of practical interest admit a particular Gaussian process (GP)representation; the GP allows us to capture the posterior with a vector of O(p)kernel hyper-parameters rather than O(p^2) interactions and main effects. Withthe implicit representation, we can run Markov chain Monte Carlo (MCMC) overmodel hyper-parameters in time and memory linear in p per iteration. We focuson sparsity-inducing models and show on datasets with a variety of covariatebehaviors that our method: (1) reduces runtime by orders of magnitude overnaive applications of MCMC, (2) provides lower Type I and Type II errorrelative to state-of-the-art LASSO-based approaches, and (3) offers improvedcomputational scaling in high dimensions relative to existing Bayesian andLASSO-based approaches.

The Kernel Interaction Trick: Fast Bayesian Discovery of Pairwise Interactions in High Dimensions