  Estimating boundary curves has many applications such as economics, climate
science, and medicine. Bayesian trend filtering has been developed as one of
locally adaptive smoothing methods to estimate the non-stationary trend of
data. This paper develops a Bayesian trend filtering for estimating the
boundary trend. To this end, the truncated multivariate normal working
likelihood and global-local shrinkage priors based on the scale mixtures of
normal distribution are introduced. In particular, well-known horseshoe prior
for difference leads to locally adaptive shrinkage estimation for boundary
trend. However, the full conditional distributions of the Gibbs sampler involve
high-dimensional truncated multivariate normal distribution. To overcome the
difficulty of sampling, an approximation of truncated multivariate normal
distribution is employed. Using the approximation, the proposed models lead to
an efficient Gibbs sampling algorithm via the P\'olya-Gamma data augmentation.
The proposed method is also extended by considering a nearly isotonic
constraint. The performance of the proposed method is illustrated through some
numerical experiments and real data examples.
