  Multidimensional scaling (MDS) is a widely used approach to representing
high-dimensional, dependent data. MDS works by assigning each observation a
location on a low-dimensional geometric manifold, with distance on the manifold
representing similarity. We propose a Bayesian approach to multidimensional
scaling when the low-dimensional manifold is hyperbolic. Using hyperbolic space
facilitates representing tree-like structures common in many settings (e.g.
text or genetic data with hierarchical structure). A Bayesian approach provides
regularization that minimizes the impact of measurement error in the observed
data and assesses uncertainty. We also propose a case-control likelihood
approximation that allows for efficient sampling from the posterior
distribution in larger data settings, reducing computational complexity from
approximately $O(n^2)$ to $O(n)$. We evaluate the proposed method against
state-of-the-art alternatives using simulations, canonical reference datasets,
Indian village network data, and human gene expression data.
