Blackwell 边界
Blackwell Boundaries
浏览论文内容
中文总结 AI 辅助
本文提出有限单纯形定理和可数诊断监控定理，刻画 Blackwell 支配，并给出三状态两信号反例，验证了相关充分性猜想的不成立。
中文摘要 AI 辅助
我们给出一个有限单纯形定理。当状态数等于源信号数且源后验似然比向量构成一个单纯形时，Blackwell 支配由一个通用的重心坐标条件刻画。我们还给出了一个显式的可数状态、可数信号的诊断监控定理，具有闭式通用准则和闭式干扰；张量化使得在每个样本量下均成立支配关系。最后，我们提供了一个三状态、两信号的反对例，针对 Mu、Pomatto、Strack 和 Tamuz 所推测的关于大样本 Blackwell 支配的多状态矩生成函数条件的充分性。该对在参数域上满足严格比较、所有有序 Kullback-Leibler 不等式、有界似然比和成对一般性，但在每个正样本量下支配均不成立。有限定理、障碍和连续不等式在 Lean 4.30 与 Mathlib 中验证；可数诊断证明以解析方式给出。
英文摘要
We give a finite simplex theorem. When the number of states equals the number of source signals and the source posterior likelihood-ratio vectors form a simplex, Blackwell dominance is characterized by a universal barycentric-coordinate condition. We also give an explicit countable-state, countable-signal diagnostic-monitor theorem with a closed-form universal criterion and a closed-form garbling; tensorization yields dominance at every sample size. Finally, we provide a three-state, two-signal counterexample to the sufficiency conjectured by Mu, Pomatto, Strack, and Tamuz for their many-state moment-generating-function conditions for large-sample Blackwell dominance. The pair satisfies strict comparisons on both parameter domains, all ordered Kullback-Leibler inequalities, bounded likelihood ratios and pairwise genericity, but dominance fails at every positive sample size. The finite theorem, obstruction, and continuum inequalities are verified in Lean 4.30 with Mathlib; the countable diagnostic proof is given analytically.
