Martingale Posterior Distributions for Log-Concave Density Functions

The family of log-concave density functions is an important class and contains various well-known probability distributions, including the normal. Due to the shape restriction, it is possible to find a nonparametric estimate of the density; the nonparametric maximum likelihood estimator (NPMLE). However, uncertainty quantification about the NPMLE via confidence bounds is less well developed. Bayesian methods are also largely absent, though it is possible to construct constrained density functions and to use 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 determine uncertainty. The algorithm can be implemented in parallel and hence is fast. We prove existence of the posterior using suitable convergence of a submartingale. We also present illustrations and comparisons with alternative approaches which include real data.