Bayesian non-linear subspace shrinkage using horseshoe priors

When modeling biological responses using Bayesian non-parametric regression, prior information may be available on the shape of the response in the form of non-linear function spaces that define the general shape of the response. To incorporate such information into the analysis, we develop a non-linear functional shrinkage (NLFS) approach that uniformly shrinks the non-parametric fitted function into a non-linear function space while allowing for fits outside of this space when the data suggest alternative shapes. This approach extends existing functional shrinkage approaches into linear subspaces to shrinkage into non-linear function spaces using a Taylor series expansion and corresponding updating of non-linear parameters. We demonstrate this general approach on the Hill model, a popular, biologically motivated model, and show that shrinkage into combined function spaces, i.e., where one has two or more non-linear functions a priori, is straightforward. We demonstrate this approach through synthetic and real data. Computational details on the underlying MCMC sampling are provided with data and analysis available in an online supplement.