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

arXiv: https://arxiv.org/abs/2111.06314 · HTML: https://mlsanity.com/paper/2111.06314/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2021.

## Built on by (most cited first, 3 of 3)

- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions](https://mlsanity.com/paper/2410.03973/) (2024)

## Abstract (arXiv)

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.