Gunn Lachlan J, Chapeau-Blondeau François, McDonnell Mark D, Davis Bruce R, Allison Andrew, Abbott Derek
School of Electrical and Electronic Engineering, The University of Adelaide , Adelaide 5005, Australia.
Laboratoire Angevin de Recherche en Ingénierie des Systèmes (LARIS) , University of Angers , 62 avenue Notre Dame du Lac, Angers 49000, France.
Proc Math Phys Eng Sci. 2016 Mar;472(2187):20150748. doi: 10.1098/rspa.2015.0748.
Is it possible for a large sequence of measurements or observations, which support a hypothesis, to counterintuitively decrease our confidence? Can unanimous support be too good to be true? The assumption of independence is often made in good faith; however, rarely is consideration given to whether a systemic failure has occurred. Taking this into account can cause certainty in a hypothesis to decrease as the evidence for it becomes apparently stronger. We perform a probabilistic Bayesian analysis of this effect with examples based on (i) archaeological evidence, (ii) weighing of legal evidence and (iii) cryptographic primality testing. In this paper, we investigate the effects of small error rates in a set of measurements or observations. We find that even with very low systemic failure rates, high confidence is surprisingly difficult to achieve; in particular, we find that certain analyses of cryptographically important numerical tests are highly optimistic, underestimating their false-negative rate by as much as a factor of 2.
一系列大量支持某一假设的测量或观察结果，有无可能违反直觉地降低我们的信心呢？一致的支持会不会好到不像是真的？独立性假设常常是善意做出的；然而，却很少有人考虑是否发生了系统性故障。考虑到这一点，可能会出现这样的情况：随着某一假设的证据明显增多，该假设的确定性却降低了。我们通过基于（i）考古证据、（ii）法律证据权衡和（iii）密码学素性测试的示例，对这种效应进行概率贝叶斯分析。在本文中，我们研究了一组测量或观察中低错误率的影响。我们发现，即使系统性故障率非常低，也极难实现高置信度；特别是，我们发现对具有密码学重要性的数值测试的某些分析非常乐观，将其假阴性率低估了多达2倍。
