  Many forecasts consist not of point predictions but concern the evolution of
quantities. For example, a central bank might predict the interest rates during
the next quarter, an epidemiologist might predict trajectories of infection
rates, a clinician might predict the behaviour of medical markers over the next
day, etc. The situation is further complicated since these forecasts sometimes
only concern the approximate "shape of the future evolution" or "order of
events". Formally, such forecasts can be seen as probability measures on spaces
of equivalence classes of paths modulo time-parametrization. We leverage the
statistical framework of proper scoring rules with classical mathematical
results to derive a principled approach to decision making with such forecasts.
In particular, we introduce notions of gradients, entropy, and divergence that
are tailor-made to respect the underlying non-Euclidean structure.
