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.