On the theory of errors and least squares
On the theory of errors and least squares
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关于误差和最小二乘理论
DOI：
10.1098/rspa.1932.0170
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发表时间：
1932
期刊：
影响因子：
--
通讯作者：
H. Jeffreys
中科院分区：
文献类型：
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作者：
H. Jeffreys
1. 在我的《科学推理》第五章中，我发现观察误差理论的通常表述需要进行一些修改，即使误差概率是按照正态分布规律分布的。所做的一项更改是精度常数 h 的先验概率的分布。虽然这通常被认为是均匀的（或被忽略），但我认为最好假设常数位于 dh 范围内的先验概率与 dh/h 成正比。这相当于假设如果 h 1/ h 2 = h 3/ h 4，则 h 可能位于 h 1 和 h 2 之间，也可能位于 h 3 和 h 4 之间；这被认为是表达先前不知道误差大小这一条件的最佳方式。对于非常小的 h（与所使用的标尺的整个长度的倒数相当）和对于大的 h（与标尺的步长的倒数相当），该关系必须破裂；但对于实际可接受的值范围来说，它似乎是最合理的分布。现在可以用另一种形式来表达对该定律的论证。正常误差定律应该成立，但真实值 x 和精度常数 h 未知。进行了两项测量：第三个观察结果之间的亲能力是多少？答案很容易看出是三分之一。
1. In my “Scientific Inference,” chapter V, I found that the usual presentation of the theory of errors of observation needed some modification, even where the probability of error is distributed according to the normal law. One change made was in the distribution of the prior probability of the precision constant h . Whereas this is usually taken as uniform (or ignored), I considered it better to assume that the prior probability that the constant lies in a range dh is proportional to dh/h . This is equivalent to assuming that if h 1/ h 2 = h 3/ h 4, h is as likely to lie between h 1 and h 2 as between h 3 and h 4; this was thought to be the best way of expressing the condition that there is no previous know ledge of the magnitude of the errors. The relation must break down for very small h , comparable with the reciprocal of the whole length of the scale used, and for large h comparable with the reciprocal of the step of the scale; but for the range of practically admissible values it appeared to be the most plausible distribution. The argument for this law can now be expressed in an alternative form. The normal law of error is supposed to hold, but the true value x and the precision constant h are unknown. Two measures are made: what is the pro ability that the third observation will lie between them ? The answer is easily seen to be one-third.