  Shape constrained regression analysis has applications in dose-response
modeling, environmental risk assessment, disease screening and many other
areas. Incorporating the shape constraints can improve estimation efficiency
and avoid implausible results. We propose two novel methods focusing on
Bayesian monotone curve and surface estimation using Gaussian process
projections. The first projects samples from an unconstrained prior, while the
second projects samples from the Gaussian process posterior. Theory is
developed on continuity of the projection, posterior consistency and rates of
contraction. The second approach is shown to have an empirical Bayes
justification and to lead to simple computation with good performance in finite
samples. Our projection approach can be applied in other constrained function
estimation problems including in multivariate settings.
