大语言模型工具中的部分相关验证级联：凹对数优势、多项式可靠性和盲点上限
Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings
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中文总结 AI 辅助
研究大语言模型工具中部分相关验证级联，将生成器错误接受率建模为潜在变量，给出理论。包括凹对数优势、多项式可靠性等特性，可测量，通过合成测试对比不同方法效果，指出实际应通过去相关而非加门提升可靠性。
中文摘要 AI 辅助
串行验证门是大语言模型工具中的核心可靠性原语：仅当k个验证器调用全部接受时才返回候选答案。在条件独立门的情况下，最近的优势定律表明后验对数优势随k线性增长，因此失败呈指数衰减，并指出“部分相关验证级联的紧密理论仍然开放”。本笔记给出了一个最小的此类理论。将生成器自身错误的每个实例错误接受率建模为潜在变量α~G（德·菲内蒂），精确的级联后验为ℓk = ℓ0 - ln mk，其中mk是G的第k阶矩。然后：（i）对于每个非退化的G，ℓk在k中是凹的——优势定律是其在第一道门处的切线和上限；（ii）对于贝塔（a，b）潜在变量，失败呈多项式衰减，1 - rk ≈ k - b，相关参数ρv = 1/（a + b + +1）；（iii）在α = 1处质量为1 - π的盲点原子将从任意数量的门中可提取的证据上限设为ln（1 - π）纳特，因此可靠性在1以下饱和；（iv）让真正接受率也变化（β~H）会产生三分法——门最终总是有帮助、平稳或积极有害——由G和H的上尾指数决定，具有封闭形式的交叉点k†。机制是幸存者偏差：通过门的错误是高α的错误。该理论是可测量的：每个实例R次重复裁决可识别G 的前R阶矩，因此两次裁决可识别ρv；贝塔 - 二项式似然和非参数最大似然估计恢复可靠性曲线和不适定上限。在合成测试中，基于独立性的外推在k = 5时将失败低估20倍，在k = 10时低估约3000倍；R = 8时的相关拟合跟踪留出的深度。实际的杠杆是去相关——改变模型族、模态或证据来源——而不是添加门。
英文摘要
Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if $k$ verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in $k$, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable $α\sim G$ (de Finetti), the exact cascade posterior is $\ell_k = \ell_0 - \ln m_k$, with $m_k$ the $k$-th moment of $G$. Then: (i) $\ell_k$ is concave in $k$ for every non-degenerate $G$ -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta$(a,b)$ latents, failure decays polynomially, $1-r_k \asymp k^{-b}$, with correlation parameter $ρ_v = 1/(a+b+1)$; (iii) a blind-spot atom of mass $1-π$ at $α=1$ caps the evidence extractable from any number of gates at $-\ln(1-π)$ nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary ($β\sim H$) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of $G$ and $H$, with closed-form crossover $k^\dagger$. The mechanism is survivorship: errors surviving gates are the high-$α$ ones. The theory is measurable: $R$ repeated verdicts per instance identify the first $R$ moments of $G$, so two verdicts identify $ρ_v$; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at $k=5$ and ~3000x at $k=10$; the correlated fit at $R=8$ tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.
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- Independent researcher（独立研究者）
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