Mathematical Sciences: Assessment of Sequential Probabilistic Forecasting Procedures

研究者考虑的问题是通过使用序列的先前值和可能的其他信息来预测随机变量序列的下一个成员。预测的形式是随机变量可能值的概率分布。（随机变量的真实分布是未知的。）评估标准或“评分规则”用以判断预测的“校准”或准确性。在预测分布正确的假设下，以及在某些方面不正确的假设下，对评分规则进行检查。研究者也将考虑这些问题的预测区域或“预测区域”，而不是一个完整的预测概率分布。（我们所说的预测区域是指在预报中从65度到73度的区间，例如：“有90%的可能性气温将在65度到73度之间。”）这些问题涉及到使用分布之间的距离度量。需要连续性论证和直接鞅论证。研究者的工作包括评估使用概率的顺序预测程序。这样的程序发生在以下例子中：气象学家给出第二天下雨的概率；医学临床医生（或专家系统）给出了特定患者患有特定疾病的主观概率；计算机专家根据过去的经验预测软件或硬件未来故障时间的分布。顺序反馈在这个过程中是必不可少的，因为当预报员知道先前预测的结果时，这一信息和其他信息被合并到下一次预测中。还将审查预测程序的长期行为。

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.