Calibration with Many Checking Rules, Mathematics of Operations Research
Each period an outcome (out of finitely many possibilities) is observed. For simplicity assume two possible outcomes, a and b. Each period, a forecaster announces the probability of a occurring next period based on the past. Consider an arbitrary subsequence of periods (e.g., odd periods, even periods, all periods in which b is observed, etc). Given an integer n, divide any such subsequence into associated sub-subsequences in which the forecast for a is between bounds. We compare the forecasts and the outcomes (realized next period) separately in each of these sub-subsequences. Given any countable partition of [0,1] and any countable collection of subsequences, we construct a forecasting scheme such that for all infinite strings of data, the long run average forecast for a matches the long run frequency of realized a's.
Rann Smorodinsky, Rakesh Vohra, Alvaro Sandroni
Smorodinsky, Rann, Rakesh Vohra, and Alvaro Sandroni. 2003. Calibration with Many Checking Rules. Mathematics of Operations Research. 28(1): 141-153.