In Hempel's paradox of the ravens, seeing a red pencil is considered as supporting evidence that all ravens are black. Also known as the Paradox of Confirmation, the paradox and its many resolutions indicate that we cannot underestimate the logical and statistical elements needed in the assessment of evidence in support of a hypothesis. Most of the previous analyses of the paradox are within the Bayesian framework. These analyses and Hempel himself generally accept the paradoxical conclusion; it feels paradoxical supposedly because the amount of evidence is extremely small. Here I describe a nonBayesian analysis of various statistical models with an accompanying likelihood-based reasoning. The analysis shows that the paradox feels paradoxical because there are natural models where observing a red pencil has no relevance to the color of ravens. In general the value of the evidence depends crucially on the sampling scheme and on the assumption about the underlying parameters of the relevant model.
翻译:在Hempel的怪兽悖论中,看到红铅笔被视为所有乌鸦都是黑色的佐证证据。又称为“确认的悖论 ”, 悖论及其许多决议表明,我们不能低估评估证据所需的逻辑和统计要素以支持假设。以前对该悖论的多数分析是在贝叶斯框架之内的。这些分析和Hempel本人一般都接受自相矛盾的结论;人们认为,由于证据数量极小,它感到自相矛盾。这里我描述了对各种统计模型的非拜耶斯人分析,并附有一种基于可能性的推理。分析表明,自相矛盾,因为有自然模型,观察红铅笔与乌鸦的颜色无关。一般而言,证据的价值取决于抽样办法和有关模型基本参数的假设。