Mathematical Sciences: Assessment of Sequential Probabilistic Forecasting Procedures

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The investigator considers the problem of predicting the next member of a sequence of random variables by using previous values of the sequence and possibly other information. The forecast takes the form of a probability distribution on the possible values of the random variable. (The true distribution of the random variable is unknown.) Assessment criteria or "scoring rules" which judge the "calibration" or accuracy of the forecast are to be examined. The scoring rules will be examined under the assumption that the forecast distribution is correct as well as under the assumption that it is incorrect in certain aspects. The investigator will also consider these problems for a forecast region or "prediction region" rather than a complete forecast probability distribution. (By a prediction region we mean the interval from 65 to 73 in a forecast such as : "There is a 90% chance that the temperature will be between 65 degrees and 73 degrees.") The problems involve using distance measures between distributions. Continguity arguments and direct martingale arguments will be needed. The investigator's work involves assessing sequential forecasting procedures that use probabilty. Such procedures occur in the following examples: a meteorologist gives a probability that it will rain the next day; medical clinicians (or expert systems) give subjective probabilities that particular patients are suffering from specific diseases; a computer specialist predicts the distribution of future failure times of software or hardware based on past experience. Sequential feedback is essential in the process in the sense that as the outcome for previous forecasts becomes known to the forecaster this and other information is incorporated into the next forecast. The long term behavior of the forecasting procedure will also be examined.