A draft of my paper “What Likelihoodists Should (and Should Not) Say About Borel’s Paradox” is here. I am not aware of any previous treatments of this issue. Addressing it is not necessary for the day-to-day use of likelihoodist methods, but it is necessary for giving a completely correct statement of the likelihoodist position.
Abstract. Borel’s paradox gives rise to counterexamples to the Law of Likelihood. Those counterexamples do not strike at the heart of the likelihoodist position, but they do require refining it. One can address them without doing substantial damage to the likelihoodist position by either restricting the Law of Likelihood so that it does not apply when the likelihood ratio in question depends on an arbitrary choice among sigma fields or maintaining that evidential favoring is relative to a sigma field in such cases. Several seemingly promising alternatives to those two responses do not work, which suggests that a likelihoodist must adopt one of them to avoid refutation.
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