Shared shrinkage horseshoe priors for Dirichlet-Tree Multinomial regression

Multivariate count data, which records counts across multiple categories within each observation, appears often in many areas of research including the physical, biological, and social sciences. Examples of these data include educational outcomes across school districts, crime rates of various types within cities, and presidential vote counts across counties. Researchers and policymakers are often interested in exploring and understanding the relationships between predictor variables like location, income, and year with the multivariate count data.

The method that we propose in this thesis incorporates a hierarchical tree structure among the categories, leveraging domain knowledge to improve the flexibility and interpretability of the model. Our method employs a shared horseshoe prior on the regression coefficients, which promotes information sharing across tree branches. In doing so, we decrease the amount of time it takes to fit a model while also improving the accuracy of the model estimates. We demonstrate these attributes through a simulation study. The process of selecting the best tree and interpreting model outputs is discussed in detail.

The benefits of the new model are showcased in the analysis of two public policy related datasets. The first study examines connections between household conditions and post-graduation intentions for high school seniors in Iowa’s public school districts. The second study explores the relationship between community characteristics and instances of various crime categories, utilizing data collected by the FBI.