Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series

Abstract

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.

Rough paths and SPDE

“…Our main result (Theorem 3.4) shows that RNN may be rewritten as a kernel method associated to the signature kernel K defined by (16). The proof is based on the neural ODE paradigm of Chen et al [32], who observed that infinite-depth residual neural networks exactly correspond to a specific type of ordinary differential equations (ODE), called neural ODE.…”