论文标题
有条件的概率和不正确的先验
Conditional probability and improper priors
论文作者
论文摘要
本文的目的是提出一种数学理论，该理论可以用作包括不当先验的统计基础。该理论包括初始公理中的不当定律，因此尤其具有贝叶斯定理。另一个结果是修改了一些通常的计算规则。这对于通常包括不当先验的常见统计实践很重要，但倾向于使用不变的计算规则。在某些情况下，结果是有效的，但在其他情况下可能会出现不一致的情况。著名的边缘化悖论体现了后一种情况。统计基础的替代数学理论可以根据条件概率空间来提出。在这种情况下，法律的出现是理论的结果。在这里证明了这两种理论的最终数学结构是等效的。结论是，最初公理的第一个或第二个公理的选择可以被视为个人喜好问题。最初对不当先验有疑问的读者可能会更开放，从有条件的概率方面来制定初始公理。不当法律的解释由相应的条件概率给出。 关键字：统计的公理，有条件的概率空间，不正确的先验，投影空间
The purpose of this paper is to present a mathematical theory that can be used as a foundation for statistics that include improper priors. This theory includes improper laws in the initial axioms and has in particular Bayes theorem as a consequence. Another consequence is that some of the usual calculation rules are modified. This is important in relation to common statistical practice which usually include improper priors, but tends to use unaltered calculation rules. In some cases the results are valid, but in other cases inconsistencies may appear. The famous marginalization paradoxes exemplify this latter case. An alternative mathematical theory for the foundations of statistics can be formulated in terms of conditional probability spaces. In this case the appearance of improper laws is a consequence of the theory. It is proved here that the resulting mathematical structures for the two theories are equivalent. The conclusion is that the choice of the first or the second formulation for the initial axioms can be considered a matter of personal preference. Readers that initially have concerns regarding improper priors can possibly be more open toward a formulation of the initial axioms in terms of conditional probabilities. The interpretation of an improper law is given by the corresponding conditional probabilities. Keywords: Axioms of statistics, Conditional probability space, Improper prior, Projective space
