Minkowski-Weyl priors for models with parameter constraints: An analysis of the BioCycle study

We propose a general framework for performing full Bayesian analysis under linear inequality parameter constraints. The proposal is motivated by the Bio Cycle Study, a large cohort study of hormone levels of healthy women where certain well-established linear inequality constraints on the log-hormone levels should be accounted for in the statistical inferential procedure. Based on the Minkowski-Weyl decomposition of polyhedral regions, we propose a class of priors that are fully supported on the parameter space with linear inequality constraints, and we fit a Bayesian linear mixed model to the Bio Cycle data using such a prior. We observe positive associations between estrogen and progesterone levels and F₂-isoprostanes, a marker for oxidative stress. These findings are of particular interest to reproductive epidemiologists.