  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.
