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.

翻译：对感兴趣函数（如回归函数或密度函数）施加形状约束（如单调性或凸性）可在无需光滑性假设的条件下实现其估计。$k$-单调性概念涵盖一族形状约束，其中递减与凸递减分别对应$k=1$和$k=2$的特例。本文考虑采用贝叶斯方法估计$k$-单调密度。通过利用核混合表示并对混合分布施加狄利克雷过程或有限混合先验，我们证明$k$-单调密度在赫林格距离下的后验收缩率为$(n/\log n)^{- k/(2k + 1)}$，该速率在多项式对数因子意义下达到极小最优。当真值$k$-单调密度为有限$J_0$成分核混合时，收缩率提升至近参数速率$\sqrt{(J_0 \log n)/n}$。进一步地，通过对$k$施加先验，我们证明即使$k$的最优值未知，上述速率仍然成立。本文还具体考虑了在大规模多重检验问题中对$p$值密度建模的应用，并通过模拟研究评估了所提方法的性能。