  This article revisits the problem of Bayesian shape-restricted inference in
the light of a recently developed approximate Gaussian process that admits an
equivalent formulation of the shape constraints in terms of the basis
coefficients. We propose a strategy to efficiently sample from the resulting
constrained posterior by absorbing a smooth relaxation of the constraint in the
likelihood and using circulant embedding techniques to sample from the
unconstrained modified prior. We additionally pay careful attention to mitigate
the computational complexity arising from updating hyperparameters within the
covariance kernel of the Gaussian process. The developed algorithm is shown to
be accurate and highly efficient in simulated and real data examples.
