  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.
