Regression analysis commonly features categorical variables and their interactions to model heterogeneous, group-specific effects. Such interactions are notorious for weak signals and cumbersome parameterizations, and thus urgently require shrinkage. However, the overwhelming default for linear models--using dummy coding, removing a reference group, and applying shrinkage--leads to unwelcome biases, asymmetries, and parameter incongruities across models. We resolve these limitations with a new parameterization-prior combination that couples abundance-based constraints for identifiability with customized shrinkage priors for categorical effects and interactions (BayesABCs). BayesABCs offers a surprising and unique invariance property: the joint posterior of all main effects is (nearly) unchanged by the addition of categorical interactions. We show how this, in conjunction with our shrinkage priors, effectively removes the principal drawbacks of including categorical interactions, thereby making heterogeneous effect estimation broadly feasible. We enforce the identifiability constraints via conditioning and provide a model representation that facilitates efficient posterior computing and direct analysis. Across simulations with a variety of categorical effects and interactions, BayesABCs consistently produces more accurate estimates and predictions, sharper inferences, and unmatched robustness to extraneous interactions relative to competing parameterizations and priors. We apply BayesABCs to study disparities in the error rates of automatic speech recognition systems.

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