Inherited Local Shrinkage for Adaptive Functional Form Discovery

Statistical models frequently employ nonlinear effects, interactions, and multiresolution basis expansions to represent complex relationships between predictors and responses. Determining the appropriate level of model complexity, however, remains a fundamental challenge. This dissertation introduces \\textit{Inherited Local Shrinkage} (ILS), a Bayesian global-local shrinkage framework for adaptive functional form discovery that enforces strong heredity among hierarchically related predictors. ILS propagates shrinkage through directed acyclic graphs representing predictor inheritance, ensuring that increasingly complex terms are retained only when supported by simpler parent structure. The framework applies broadly to polynomial regression, interaction models, wavelets, splines, and other hierarchical basis expansions, while naturally extending to adaptive multiresolution modeling in which the level of spatial or temporal detail is learned from the data. The methodology is applied to two substantive environmental and ecological problems: identifying drivers of indoor air quality across North America and modeling freshwater turtle occupancy in northwest Oregon under imperfect detection. Together, these methodological developments and applications demonstrate that inherited local shrinkage provides a general and computationally practical framework for adaptive functional form discovery across a broad class of Bayesian regression models.