发表机构
The Johns Hopkins University; The City College of New York & The Graduate Center, The City University of New York; The University of Texas at Austin（约翰斯·霍普金斯大学; 纽约市立学院及纽约市立大学研究生院; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文指出Nielsen关于贝叶斯收敛定理的证明错误，构造反例表明两个性质非充分，并修正了部分结果，揭示可数可加性并非充分条件。
AI 中文摘要
我们重新审视了Nielsen最近一篇论文（J. Philos. Logic, 2021, doi: https://doi.org/10.1007/s10992-020-09569-2）中关于有限可加性下贝叶斯收敛于真理的分析。其主要定理证明，一个概率函数的后验几乎一致地收敛于真理，当且仅当该函数具有两个性质：一个逼近性质，以及所谓的条件命中时间上的可数可加性。我们证明这两个性质是必要的但非充分的，因此该定理是错误的，并定位了其已发表证明中的错误。我们构造了一个仅有限可加的概率函数，该函数具有这两个性质，且其后验几乎必然收敛于真理但并非几乎一致收敛。该论文四个剩余定理中的三个及其推论也失去了其已发表的证明，其中两个因失败的蕴含关系，另两个因我们同样识别出的一个单独缺陷。四个结果中我们重新证明了两个，一个我们留作未决；最后一个是推论，我们的反例并未反驳它，并且我们为一个包含该反例的族建立了该推论，但在一般情况下留作开放。我们还证明，对每个事件，后验几乎必然收敛于真理并不刻画可数可加性。我们以修复后的几乎一致收敛刻画所需补充的内容作为结尾。
英文摘要
We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.
