Calibration with Many Checking Rules

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Abstract
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 subsubsequences in which the forecast for a is between [i /n, i+1/n), i ∈ {0, 1, …, n}. We compare the forecasts and the outcomes (realized next period) separately in each of these subsubsequences. 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. A good Christian must beware of mathematicians and those soothsayers who make predictions by unholy methods, especially when their predictions come true, lest they ensnare the soul through association with demons. St. Augustine, De genesis ad litteram, Book II.

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