Low-rank matrix completion methods have been successful in a variety of settings such as recommendation systems. However, most of the existing matrix completion methods only provide a point estimate of missing entries, and do not characterize uncertainties of the predictions. In this paper, we propose a Bayesian hierarchical probabilistic matrix factorization (BHPMF) model to 1) incorporate hierarchical side information, and 2) provide uncertainty quantified predictions. The former yields significant performance improvements in the problem of plant trait prediction, a key problem in ecology, by leveraging the taxonomic hierarchy in the plant kingdom. The latter is helpful in identifying predictions of low confidence which can in turn be used to guide field work for data collection efforts. A Gibbs sampler is designed for inference in the model. Further, we propose a multiple inheritance BHPMF (MI-BHPMF) which can work with a general directed acyclic graph (DAG) structured hierarchy, rather than a tree. We present comprehensive experimental results on the problem of plant trait prediction using the largest database of plant traits, where BHPMF shows strong empirical performance in uncertainty quantified trait prediction, outperforming the state-of-the-art based on point estimates. Further, we show that BHPMF is more accurate when it is confident, whereas the error is high when the uncertainty is high.