  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 $(n/\log n)^{- k/(2k + 1)}$ for a $k$-monotone density,
which is minimax optimal up to a polylogarithmic factor. When the true
$k$-monotone density is a finite $J_0$-component mixture of the kernel, the
contraction rate improves to the nearly parametric rate $\sqrt{(J_0 \log
n)/n}$. 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.
