# Probability Paths and the Structure of Predictions over Time

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.