  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.
