Shared Bayesian variable shrinkage in multinomial logistic regression

Multiple Bayesian approaches have been explored for variable selection in the multinomial regression framework. While there are a number of studies considering variable selection in the regression paradigm with a numerical response, the research is limited for a categorical response variable. The proposed approach develops a method for leveraging the features of the global-local shrinkage framework to improve variable selection in baseline categorical logistic regression by introducing new shrinkage priors that encourage similar predictors to be selected across the models for different response levels. To that end, the proposed shrinkage priors share information across response models through the local parameters that favor similar levels of shrinkage for all coefficients (log-odds ratios) of a predictor. Different shrinkage approaches are explored using the horseshoe and normal gamma priors within this setting and compared to a spike-and-slab setup and other shrinkage priors that fail to share information across models. The performance of the approach is investigated in both simulations and a real data application.