The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.
翻译:Anna Karenina 原则是在匿名小说中的开篇句后命名的: 幸福家庭都是一样; 每个不幸福的家庭都是一样; 每个不幸福的家庭都以自己的方式不幸福。 两个百分百问题(TEP)是一个在概率理论、数学经济学、逻辑和哲学方面受到广泛研究的悖论。 一次又一次地发表新的分析, 作者声称最终要解释这个悖论中到底出了什么问题。 每个作者( 包括目前的作者) 都强调他们的方法中的新颖之处, 并得出结论, 早期的方法并没有找到问题的根源。 我们发现, 虽然逻辑论只是每个步骤都是正确的, 一个显然合乎逻辑的争论才是正确的。 一个显然不合逻辑的争论可以被视为在不同的地方走迷路。 这导致将TEP的文献和成功的电影专栏进行对比: 它产生一系列的续集, 甚至是预言, 每个不同的导演都以个人的方式对待同样的基本前提。 我们用文献中的决议来综合,纠正常见的错误, 并且给出一个新的理论, 一个可以互换的变量的顺序属性, 一个显然会被视为在不同地方上走得更深层次的变式的变式, 。