A nonBayesian view of Hempel's paradox of the ravens

TL;DR

This paper offers a non-Bayesian, likelihood-based analysis of Hempel's paradox of the ravens, emphasizing the importance of sampling schemes and model assumptions in evaluating evidence.

Contribution

It introduces a non-Bayesian approach to the paradox, highlighting how different models and sampling schemes influence the perceived support from evidence.

Findings

Certain models show no relevance of observing a red pencil to raven color

Evidence value depends on sampling scheme and model assumptions

Paradoxical conclusions are model-dependent

Abstract

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.