With the explosive growth of machine learning techniques and applications, new paradigms and models with transformative power are enriching the field. One of the most remarkable trends in recent years is the rapid rise of significance of Riemannian geometry and Lie group theory. The underlying cause is the rising complexity of the data, motivating more sophisticated approaches, thus leading to the wide recognition that a great deal of data sets exhibit an intrinsic curvature. In other words, many data sets are naturally represented or faithfully embedded into non-Euclidean spaces. One apparent example of this kind are rotational motions in robotics.

We introduce the novel family of probability distributions on hyperbolic disc. The distinctive property of the proposed family is invariance under the actions of the group of disc-preserving conformal mappings. The group-invariance property renders it a convenient and tractable model for encoding uncertainties in hyperbolic data. Potential applications in Geometric Deep Learning and bioinformatics are numerous, some of them are briefly discussed. We also emphasize analogies with hyperbolic coherent states in quantum physics.