Hidden mathematical structure may be used to reveal hidden biological structure. These images are low dimensional projections of samples from a high dimensional probability distribution. Exploring such distributions is vital for Bayesian statistical inference of mechanistic biological models, which can shed light on the most likely hidden physical structure of a biological system, such as a cell signalling network, based on limited experimental observations. The left hand image shows a random walk through the probability space of a mechanistic biological model, with the red lines representing rejected moves and hence wasted computational effort. The right hand image utilises an additional hidden geometric structure; by representing the probability space as a Riemannian manifold, moves can be made along "geodesics" or shortest paths between two points. The resulting exploration is orders of magnitude faster than a random walk, and makes possible probabilistic ranking of complex biological hypotheses.