  The output of predictive models is routinely recalibrated by reconciling
low-level predictions with known derived quantities defined at higher levels of
aggregation. For example, models predicting turnout probabilities at the
individual level in U.S. elections can be adjusted so that their aggregation
matches the observed vote totals in each state, thus producing better
calibrated predictions. In this research note, we provide theoretical grounding
for one of the most commonly used recalibration strategies, known colloquially
as the "logit shift." Typically cast as a heuristic optimization problem
(whereby an adjustment is found such that it minimizes the difference between
aggregated predictions and the target totals), we show that the logit shift in
fact offers a fast and accurate approximation to a principled, but often
computationally impractical adjustment strategy: computing the posterior
prediction probabilities, conditional on the target totals. After deriving
analytical bounds on the quality of the approximation, we illustrate the
accuracy of the approach using Monte Carlo simulations. The simulations also
confirm analytical results regarding scenarios in which users of the simple
logit shift can expect it to perform best -- namely, when the aggregated
targets are comprised of many individual predictions, and when the distribution
of true probabilities is symmetric and tight around 0.5.
