Shape restriction, like monotonicity or convexity, imposed on a function of
interest, such as a regression or density function, allows for its estimation
without smoothness assumptions. The concept of k-monotonicity encompasses a
family of shape restrictions, including decreasing and convex decreasing as
special cases corresponding to k=1 and k=2. We consider Bayesian approaches
to estimate a k-monotone density. By utilizing a kernel mixture
representation and putting a Dirichlet process or a finite mixture prior on the
mixing distribution, we show that the posterior contraction rate in the
Hellinger distance is (n/logn)βk/(2k+1) for a k-monotone density,
which is minimax optimal up to a polylogarithmic factor. When the true
k-monotone density is a finite J0β-component mixture of the kernel, the
contraction rate improves to the nearly parametric rate (J0βlogn)/nβ. Moreover, by putting a prior on k, we show that the same rates hold
even when the best value of k is unknown. A specific application in modeling
the density of p-values in a large-scale multiple testing problem is
considered. Simulation studies are conducted to evaluate the performance of the
proposed method