We consider a nonparametric Bayesian approach to estimation and testing for a multivariate monotone density. Instead of following the conventional Bayesian route of putting a prior distribution complying with the monotonicity restriction, we put a prior on the step heights through binning and a Dirichlet distribution. An arbitrary piece-wise constant probability density is converted to a monotone one by a projection map, taking its $\mathbb{L}_1$-projection onto the space of monotone functions, which is subsequently normalized to integrate to one. We construct consistent Bayesian tests to test multivariate monotonicity of a probability density based on the $\mathbb{L}_1$-distance to the class of monotone functions. The test is shown to have a size going to zero and high power against alternatives sufficiently separated from the null hypothesis. To obtain a Bayesian credible interval for the value of the density function at an interior point with guaranteed asymptotic frequentist coverage, we consider a posterior quantile interval of an induced map transforming the function value to its value optimized over certain blocks. The limiting coverage is explicitly calculated and is seen to be higher than the credibility level used in the construction. By exploring the asymptotic relationship between the coverage and the credibility, we show that a desired asymptomatic coverage can be obtained exactly by starting with an appropriate credibility level.

翻译：本文探讨了多元单调密度估计与检验的非参数贝叶斯方法。我们不采用传统贝叶斯路径（即施加满足单调性约束的先验分布），而是通过分箱和狄利克雷分布为阶梯高度设定先验。通过投影映射将任意分段常数概率密度转化为单调密度——即取其到单调函数空间的$\mathbb{L}_1$投影，随后对结果归一化使其积分值为1。我们构建了相合的贝叶斯检验，基于概率密度到单调函数类的$\mathbb{L}_1$距离检验其多元单调性。该检验的尺度趋近于零，且对于与零假设充分分离的备择假设具有高功效。为获取密度函数内点值的贝叶斯可信区间（保证渐近频率覆盖），我们考虑通过将函数值优化至特定区块上的诱导映射的后验分位数区间。极限覆盖率的显式表达式表明其高于构造中使用的可信水平。通过探究覆盖率与可信水平之间的渐近关系，我们证明从适当的可信水平出发，可精确实现期望的渐近覆盖率。