11. Nonparametric Bayesian Regression Under Combinations of Local Shape Constraints

A nonparametric Bayesian method for regression under combinations of local shape constraints is proposed. The shape constraints considered include monotonicity, concavity (or convexity), unimodality, and in particular, combinations of several types of range-restricted constraints. By using a B-spline basis, the support of the prior distribution is included in the set of piecewise polynomial functions. The first novelty is that, thanks to the local support property of B-splines, many combinations of constraints can easily be considered by identifying B-splines whose support intersects with each constrained region. Shape constraints are included in the coefficients prior using a truncated Gaussian distribution. However, the prior density is only known up to the normalizing constant, which does change with the dimension of coefficients. The second novelty is that we propose to simulate from the posterior distribution by using a reversible jump MCMC slice sampler, for selecting the number and the position of the knots, coupled to a simulated annealing step to project the coefficients on the constrained space. This method is valid for any combination of local shape constraints and particular attention is paid to the construction of a trans-dimensional MCMC scheme.