# Threshold Martingales and the Evolution of Forecasts

arXiv: https://arxiv.org/abs/2105.06834 · HTML: https://mlsanity.com/paper/2105.06834/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 10 references that are in the map.
- First posted on arXiv 14 May 2021.

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

- [Learning an Inventory Control Policy with General Inventory Arrival Dynamics](https://mlsanity.com/paper/2310.17168/) (2023)
- [Probability Paths and the Structure of Predictions over Time](https://mlsanity.com/paper/2106.06515/) (2021)
- [Forking-Sequences: Statistically and Computationally Efficient Multi-Horizon Forecasting with Reduced Volatility](https://mlsanity.com/paper/2510.04487/) (2025)

## Abstract (arXiv)

This paper introduces a martingale that characterizes two properties of evolving forecast distributions. Ideal forecasts of a future event behave as martingales, sequen- tially updating the forecast to leverage the available information as the future event approaches. The threshold martingale introduced here measures the proportion of the forecast distribution lying below a threshold. In addition to being calibrated, a threshold martingale has quadratic variation that accumulates to a total determined by a quantile of the initial forecast distribution. Deviations from calibration or to- tal volatility signal problems in the underlying model. Calibration adjustments are well-known, and we augment these by introducing a martingale filter that improves volatility while guaranteeing smaller mean squared error. Thus, post-processing can rectify problems with calibration and volatility without revisiting the original forecast- ing model. We apply threshold martingales first to forecasts from simulated models and then to models that predict the winner in professional basketball games.