一种新的改进贝叶斯方法用于测量不确定度分析及频率与贝叶斯推理的统一

Hening Huang
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引用次数: 1

摘要

本文提出了一种对传统贝叶斯测量不确定度分析方法的改进。新的改进贝叶斯方法是根据信息聚集规律和频率论观点与贝叶斯观点之间的转换规律推导出来的。它也可以由原贝叶斯定理导出为连续形式。我们关注测量科学中经常遇到的一个问题:一个测量给出了一系列的观察结果。我们考虑两种情况:(1)测量量没有真正的先验信息,因此不确定性评估纯粹是A型;(2)先验信息是可用的,并且由正态分布表示。传统的贝叶斯方法(也称为重新表述的贝叶斯定理)在这两种情况下都不能提供有效的标准不确定度估计。新的改进贝叶斯方法对这两种情况提供了与频率论相同的解决方案。讨论了新改进的贝叶斯方法与传统贝叶斯方法的区别。本文揭示了传统的贝叶斯方法不是一个自洽的操作,因此在某些情况下可能会导致错误的推断,例如所考虑的两种情况。根据频率-贝叶斯变换规则和信息聚集定律(LAI),频率和贝叶斯推理实际上是等价的,因此至少在测量不确定度分析中是可以统一的。这种统一引起了相当大的兴趣,因为它可能解决频率论者和贝叶斯论者之间长期存在的争论。统一还可能导致对GUM(测量数据的评价-测量不确定度表达指南(JCGM 2008))进行无可争议的统一修订。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A new modified Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference
This paper proposes a new modification of the traditional Bayesian method for measurement uncertainty analysis.  The new modified Bayesian method is derived from the law of aggregation of information (LAI) and the rule of transformation between the frequentist view and Bayesian view.  It can also be derived from the original Bayes Theorem in continuous form.  We focus on a problem that is often encountered in measurement science: a measurement gives a series of observations.  We consider two cases: (1) there is no genuine prior information about the measurand, so the uncertainty evaluation is purely Type A, and (2) prior information is available and is represented by a normal distribution.  The traditional Bayesian method (also known as the reformulated Bayes Theorem) fails to provide a valid estimate of standard uncertainty in either case.  The new modified Bayesian method provides the same solutions to these two cases as its frequentist counterparts.  The differences between the new modified Bayesian method and the traditional Bayesian method are discussed.  This paper reveals that the traditional Bayesian method is not a self-consistent operation, so it may lead to incorrect inferences in some cases, such as the two cases considered.  In the light of the frequentist-Bayesian transformation rule and the law of aggregation of information (LAI), the frequentist and Bayesian inference are virtually equivalent, so they can be unified, at least in measurement uncertainty analysis.  The unification is of considerable interest because it may resolve the long-standing debate between frequentists and Bayesians.  The unification may also lead to an indisputable, uniform revision of the GUM (Evaluation of measurement data - Guide to the expression of uncertainty in measurement (JCGM 2008)).
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