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Uninformative Priors for Bayes' Theorem

机译：贝叶斯定理的非先验先验

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One of the major objections to the uses of Bayesian Methods and Maximum Entropy Methods is associated with the choice of the prior distribution, especially when there is no prior information available. LaPlace, who made excellent use of Bayesian methods to obtain significant results, originally postulated that the uninformative prior should be constant, although he later came to question the use of this prior. One of the objections to the use of a constant prior is that it is not independent of variable transformations: the thought being that if a prior is truly uninformative, then any transformation of the variable should also be uninformative. Later, Jeffreys, using heuristic arguments, postulated that a better uninformative prior is the 1/σfunction. Jaynes provided a more formal derivation of the 1/σ prior for certain classes of probability functions. The 1/σ prior has the desirable feature of being independent of integer-power variable transformations, although it is not a probability distribution, and is therefore called an improper prior. Unfortunately, the 1/σ prior gives extreme weighting to low values of σ. Shannon, while working at Bell Telephone Laboratories, derived Statistical Entropy, as a measure of uncertainty in a distribution. Shannon's work leads to a constant prior, in the uninformative case. Unfortunately, Shannon's Entropy holds only for discrete variable distributions. In this paper, an uninformative prior in the continuous variable case is developed, which is independent of virtually any variable transformation and yet does not give preferential weight to any value of the continuous variable. A simple example is used to show how the results differ for the new prior and the 1/σ prior.
机译：使用贝叶斯方法和最大熵方法的主要反对意见之一与先验分布的选择有关，尤其是在没有先验信息可用的情况下。拉普拉斯（LaPlace）善用贝叶斯方法来获得显着结果，他最初假设无信息的先验应该是恒定的，尽管后来他开始质疑先验的使用。使用常量先验的一个反对意见是，它不独立于变量转换：认为如果先验确实是无意义的，那么对变量的任何转换也应是无信息的。后来，杰弗里斯（Jeffreys）使用启发式的论证假设1 /σ函数是一个更好的无信息先验。 Jaynes为某些类别的概率函数提供了1 /σ先验的更正式形式。 1 /σ先验具有独立于整数幂变量变换的期望特征，尽管它不是概率分布，因此被称为不正确先验。不幸的是，1 /σ先验值给σ的低值赋予了极大的权重。 Shannon在贝尔电话实验室工作时，推导了统计熵，以度量分布中的不确定性。在没有信息的情况下，香农的工作导致了一个恒定的先验。不幸的是，香农熵仅适用于离散变量分布。在本文中，开发了一种在连续变量情况下无信息的先验，它实际上与任何变量转换无关，但并未对连续变量的任何值赋予优先权。一个简单的例子用来说明新先验和1 /σ先验的结果如何不同。

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