Probability Paths and the Structure of Predictions over Time

TL;DR

This paper introduces the Gaussian latent information martingale (GLIM), a Bayesian framework for modeling the evolution of probability forecasts over time, capturing their structure and uncertainty more accurately than existing methods.

Contribution

The paper presents GLIM, a novel Bayesian approach that models the dynamic structure of probability paths, preserving martingale properties and improving uncertainty quantification.

Findings

GLIM outperforms three baseline methods in estimating probability path distributions.

The approach effectively captures the volatility and information flow in probability forecasts.

GLIM enhances decision-making by better modeling the evolution of predictions over time.

Abstract

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.