# Probability Paths and the Structure of Predictions over Time

arXiv: https://arxiv.org/abs/2106.06515 · HTML: https://mlsanity.com/paper/2106.06515/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 73 references that are in the map.
- First posted on arXiv 11 Jun 2021.

## Builds on (most cited first, 10 of 10)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Fair prediction with disparate impact: A study of bias in recidivism prediction instruments](https://mlsanity.com/paper/1610.07524/) (2016)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [A Survey of Inverse Reinforcement Learning: Challenges, Methods and Progress](https://mlsanity.com/paper/1806.06877/) (2018)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [Bayesian Convolutional Neural Networks with Bernoulli Approximate Variational Inference](https://mlsanity.com/paper/1506.02158/) (2015)
- [MQTransformer: Multi-Horizon Forecasts with Context Dependent and Feedback-Aware Attention](https://mlsanity.com/paper/2009.14799/) (2020)
- [Threshold Martingales and the Evolution of Forecasts](https://mlsanity.com/paper/2105.06834/) (2021)

## Built on by (most cited first, 1 of 1)

- [Robust Predictions with Ambiguous Time Delays: A Bootstrap Strategy](https://mlsanity.com/paper/2408.12801/) (2024)

## Abstract (arXiv)

In settings ranging from weather forecasts to political prognostications to financial projections, probability estimates of future binary outcomes often evolve over time. For example, the estimated likelihood of rain on a specific day changes by the hour as new information becomes available. Given a collection of such probability paths, we introduce a Bayesian framework -- which we call the Gaussian latent information martingale, or GLIM -- for modeling the structure of dynamic predictions over time. Suppose, for example, that the likelihood of rain in a week is 50 %, and consider two hypothetical scenarios. In the first, one expects the forecast to be equally likely to become either 25 % or 75 % tomorrow; in the second, one expects the forecast to stay constant for the next several days. A time-sensitive decision-maker might select a course of action immediately in the latter scenario, but may postpone their decision in the former, knowing that new information is imminent. We model these trajectories by assuming predictions update according to a latent process of information flow, which is inferred from historical data. In contrast to general methods for time series analysis, this approach preserves important properties of probability paths such as the martingale structure and appropriate amount of volatility and better quantifies future uncertainties around probability paths. We show that GLIM outperforms three popular baseline methods, producing better estimated posterior probability path distributions measured by three different metrics. By elucidating the dynamic structure of predictions over time, we hope to help individuals make more informed choices.