Bayesian nonparametric estimation in the current status continuous mark model

TL;DR

This paper introduces a Bayesian nonparametric approach for estimating the joint distribution of event times and marks in a current status model, providing theoretical contraction rates and computational methods.

Contribution

It develops a novel Bayesian nonparametric method with specific priors for the current status continuous mark model, including theoretical contraction rates and practical sampling algorithms.

Findings

Posterior distribution contracts at rate (n/log n)^{-ρ/3(ρ+2)}.

Method performs well in simulated examples.

Provides computational algorithms for posterior sampling.

Abstract

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.