The family of log-concave density functions contains various kinds of common probability distributions. Due to the shape restriction, it is possible to find the nonparametric estimate of the density, for example, the nonparametric maximum likelihood estimate (NPMLE). However, the associated uncertainty quantification of the NPMLE is less well developed. The current techniques for uncertainty quantification are Bayesian, using a Dirichlet process prior combined with the use of Markov chain Monte Carlo (MCMC) to sample from the posterior. In this paper, we start with the NPMLE and use a version of the martingale posterior distribution to establish uncertainty about the NPMLE. The algorithm can be implemented in parallel and hence is fast. We prove the convergence of the algorithm by constructing suitable submartingales. We also illustrate results with different models and settings and some real data, and compare our method with that within the literature.

翻译：对数凹密度函数族包含多种常见概率分布。由于形状约束的限制，可以找到密度的非参数估计，例如非参数最大似然估计（NPMLE）。然而，NPMLE的相关不确定性量化尚不完善。当前的不确定性量化技术基于贝叶斯方法，使用狄利克雷过程先验并结合马尔可夫链蒙特卡洛（MCMC）从后验分布中进行采样。本文从NPMLE出发，利用鞅后验分布的一种变体来建立关于NPMLE的不确定性。该算法可并行实现，因此计算速度快。我们通过构造适当的子鞅证明了算法的收敛性。同时，我们针对不同模型、设置及实际数据展示了结果，并与文献中的方法进行了比较。