Bayesian nonparametric estimation in the current status continuous mark model

In this paper we consider the current status continuous mark model where, if the event takes place before an inspection time $T$ a "continuous mark" variable is observed as well. A Bayesian nonparametric method is introduced for estimating the distribution function of the joint distribution of the event time ($X$) and mark ($Y$). We consider a prior that is obtained by assigning a distribution on heights of cells, where cells are obtained from a partition of the support of the density of $(X, Y)$. As distribution on cell heights, we consider both a Dirichlet prior and a prior based on the graph-Laplacian on the specified partition. Our main result shows that under appropriate conditions, the posterior distribution function contracts pointwisely at rate $\left(n/\log n\right)^{-\frac{\rho}{3(\rho+2)}}$, where $\rho$ is the H\"older smoothness of the true density. In addition to our theoretical results, we provide computational methods for drawing from the posterior using probabilistic programming. The performance of our computational methods is illustrated in two examples.