Research on an Objective Ensemble-Oriented Theory of Lower Probability

自17世纪以来，传统的概率论在描述偶然性、不确定性和不确定性现象方面取得了巨大成功。它的成功是如此巨大，以至于人们对开发替代传统位置的产品几乎没有兴趣。费恩教授和他的学生们正在继续他们对另一种选择的结构、解释和应用的长期研究，即低概率的频率论理论，他们认为这种理论比传统方法有相当大的优势。他们所说的“低概率”指的是类概率但超加性非负和单位形成的集合函数，其中的例子包括熟悉的可数和有限加性概率测度，由Demster和Shafer开创的广泛讨论的信念函数，Kyburg, Levi, Suppes， Walley等人研究的低包膜，以及由法恩教授和他的学生引入的非占多数低概率类。在这项续期拨款下，他们的目标是发展基于低概率的推理，以研究不确定的物理和自然现象，特别是发展和研究一种统计理论，该理论可以将时间序列数据与统计理论提供的低概率模型相协调。他们期望利用他们最近发展的低期望的定义作为估计理论和通过条件反射从观察中学习的基础。虽然其他人提供的较低概率的解释要么是主观的，要么是认识论的，但他们的努力将继续集中在客观频率相关解释的可能性上。这种对低概率的客观解释的研究也应该增强我们对常规概率的这种解释的理解。

Conventional probability theory has, since the 17th century, been enormously successful in characterizing chance, uncertainty, and indeterminate phenomena. Its success has been so great that there has been little interest in developing alternatives to the conventional position. Professor Fine and his students are continuing their long-term study of the structure, interpretation and application of one alternative, the frequentist theory of lower probability, which they believe offers considerable advantages over the traditional approach. By "lower probability" they refer to a probability-like but superadditive nonnegative and unit formed set function, examples of which include the familiar countably and finitely additive probability measures, the widely discussed belief functions pioneered by Demster and Shafer, the lower envelopes studied by Kyburg, Levi, Suppes, Walley and others, and the class of undominated lower probabilities introduced by Professor Fine and his students. Under this renewal grant, they aim to develop lower probability- based reasoning about nondeterministic physical and natural phenomena and particularly the development and study of a theory of statistics that can coordinate time series data with the lower probability models that are provided by the statistical theory. They expect to exploit their recently developed definition of lower expectation as a basis for a theory of estimation and for learning from observations through conditioning. While the interpretations of lower probability offered by others have been of either subjective or epistemic varieties, their efforts will continue to concentrate on the possibilities of an objective frequentist-related interpretation. This research on an objective interpretation of lower probability should also enhance our understanding of such interpretations for conventional probability.