Statistics has made tremendous advances since the times of Fisher, Neyman, Jeffreys, and others, but the fundamental questions about probability and inference that puzzled our founding fathers still exist and might even be more relevant today. To overcome these challenges, I propose to look beyond the two dominating schools of thought and ask what do scientists need out of statistics, do the existing frameworks meet these needs, and, if not, how to fill the void? To the first question, I contend that scientists seek to convert their data, posited statistical model, etc., into calibrated degrees of belief about quantities of interest. To the second question, I argue that any framework that returns additive beliefs, i.e., probabilities, necessarily suffers from false confidence---certain false hypotheses tend to be assigned high probability---and, therefore, risks making systematically misleading conclusions. This reveals the fundamental importance of non-additive beliefs in the context of statistical inference. But non-additivity alone is not enough so, to the third question, I offer a sufficient condition, called validity, for avoiding false confidence, and present a framework, based on random sets and belief functions, that provably meets this condition.
翻译:自费舍尔、尼曼、杰弗里斯等时代以来,统计取得了巨大进步,但令我们的创始人感到困惑的概率和推论的根本问题仍然存在,而且今天甚至可能更加相关。为了克服这些挑战,我提议超越这两个主导思想流派,并询问科学家需要从统计中拿出什么数据,现有框架是否满足这些需要,如果不是,如何填补空白?关于第一个问题,我认为科学家们试图将数据、统计模型等转换成对利益数量的一致信仰程度。关于第二个问题,我认为,任何返回累加信念的框架,即概率,必然受到虚假信心-不确定性的困扰,因此,有可能系统地作出误导性结论。这显示了不增加信仰在统计推论中的根本重要性。但仅不增加信仰本身还不够,对于第三个问题来说,我提供了充分的条件,要求有效性,以避免虚假信心,并基于随机设定和假设的功能,提出框架。