Is Bayes Posterior just Quick and Dirty Confidence?
Is Bayes Posterior just Quick and Dirty Confidence?
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DOI：
10.1214/11-sts352
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发表时间：
2011-08-01
影响因子：
5.7
通讯作者：
Fraser, D. A. S.
中科院分区：
文献类型：
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作者：
Fraser, D. A. S.
贝叶斯[Philos. Soc. Lond. 53（1763）370-418; 54 296-325]将观测到的似然函数引入到统计推断中，并提供了一个权重函数来校准参数;他还引入了参数空间上的置信分布，但没有提供目前的理由。当然，可能性和信心的名字直到很久以后才出现：Soc. Lond.数学、物理、工程、科学系A。222（1922）309-368]的可能性和奈曼[Philos.Trans.R. Soc. Lond.数学、物理、工程、科学系A。237（1937）333-380]的信心。林德利[J.罗伊。中央集权主义者Soc. Ser. B 20（1958）102-107]表明，当模型不是位置时，贝叶斯和置信度结果是不同的。本文探讨了从贝叶斯方法和置信度方法的真实陈述的发生，并表明，在贝叶斯情况下，真实陈述的比例严重依赖于模型中线性的存在;偏离这种线性，贝叶斯方法可能是一个很差的近似值，并严重误导。因此，加权似然的贝叶斯积分提供了置信度的一阶线性近似，但如果没有线性，可能会给出实质上不正确的结果。
Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370-418; 54 296-325] introduced the observed likelihood function to statistical inference and provided a weight function to calibrate the parameter; he also introduced a confidence distribution on the parameter space but did not provide present justifications. Of course the names likelihood and confidence did not appear until much later: Fisher [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 222 (1922) 309-368] for likelihood and Neyman [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 237 (1937) 333-380] for confidence. Lindley [J. Roy. Statist. Soc. Ser. B 20 (1958) 102-107] showed that the Bayes and the confidence results were different when the model was not location. This paper examines the occurrence of true statements from the Bayes approach and from the confidence approach, and shows that the proportion of true statements in the Bayes case depends critically on the presence of linearity in the model; and with departure from this linearity the Bayes approach can be a poor approximation and be seriously misleading. Bayesian integration of weighted likelihood thus provides a first-order linear approximation to confidence, but without linearity can give substantially incorrect results.