Shape restrictions such as monotonicity in one or more dimensions sometimes naturally arise. The restriction can be effectively used for function estimation without smoothing. Several exciting results on function estimation under monotonicity, and to a lesser extent, under multivariate monotonicity have been obtained in the frequentist setting. But only a little is known about how Bayesian methods work when there are restrictions on the shape. Chakraborty and Ghosal recently studied the convergence properties of a "projection-posterior" distribution. The shape restriction is not imposed on the prior in this approach. Instead, a conjugate prior disregarding the shape is used. Samples from the posterior distribution are "corrected" via a projection map to comply with the shape restriction. It was found that the equal-tailed projection-posterior credible interval for the function value at a point has a limiting coverage slightly higher than the credibility, which is the opposite of the phenomenon observed by Cox for smooth functions. Interestingly, the correct coverage is obtained for a suitably lower credibility interval. In the multivariate context, we generalize the projection-posterior approach by using an ``immersion map'' given by a block maxmin operation and show that the resulting Bayesian credible intervals have similar coverage properties by explicitly evaluating the limiting coverage in terms of a function of a pair of Gaussian processes. Simulation results confirm the theoretical findings.

This talk is based on joint work with Kang Wang, a doctoral student of the speaker at North Carolina State University.