发表机构
Hunter College and the Graduate Center, CUNY（亨特学院与CUNY研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用单一贝叶斯生成模型在四个节点（似然、模型类、群体/情境依赖、报告）的失败，统一解释疼痛定位的多种失败模式，并给出可估计的空间模型与实验设计。
AI 中文摘要
患者报告的疼痛定位对某些临床表现具有诊断决定性，而对其他情况几乎无信息价值。一篇配套论文认为，这并非诊断效用的单一梯度，而是定位失败的三种不同形式。本文为这些失败提供了数学描述，并表明它们是同一个对象：一个单一的贝叶斯生成模型在不同节点——似然、模型类别以及该似然中的群体或情境依赖性——发生失败。这一计数并无争议：前三种是产生感觉定位的推断失败，而第四种（此处新增）是报告该定位的失败。解剖学多路复用是一个不可辨识的反问题：一个秩亏的转介矩阵将不同的病因映射到同一报告。去局部化放大是生成模型的改变，其动力学为神经场分岔，以空间范围为序参量。转介和非典型位移是呈现似然中的群体或情境依赖性——这是一个概率，而非损失——其后果位于下游一层，即随群体患病率和成本变化的决策阈值。重复观察无法减少关于固定推断目标的可恢复信息，这是一个精确的链式法则恒等式，而实际价值可能下降。第四个节点是报告本身。一个空间贝叶斯模型将错误定位分解为转介模糊、解剖偏移和精度加权的认知覆盖，并将幻肢、镜像盒和中枢卒中后报告重现为同一方程的不同状态。该模型在论文陈述为必要或充分而非假设的条件下是可估计的；第8节给出了满足这些条件的设计。一个对照被报告为与论文自身利益相反：首次用于测量迁移的传输统计量将一个静止的、扩散的轮廓评分得仿佛它已经移动了。
英文摘要
Patient-reported pain location is diagnostically decisive for some presentations and nearly uninformative for others. A companion paper argues this is not one gradient of diagnostic utility but three distinct failures of localization. This paper gives those failures their mathematics and shows they are one object: a single Bayesian generative model failing at different nodes - the likelihood, the model class, and group- or context-dependence in that same likelihood. The count is not in dispute: the first three are failures of the inference that produces a felt location, and the fourth, added here, is a failure of reporting it. Anatomical multiplexing is a non-identifiable inverse problem: a rank-deficient referral matrix sends distinct causes to one report. Delocalized amplification is a change of generative model whose dynamics are a neural-field bifurcation, with spatial extent as the order parameter. Referred and atypical displacement is group- or context-dependence in the presentation likelihood - a probability, not a loss - whose consequence sits one layer downstream, in a decision threshold varying with group prevalence and cost. Repeated observation cannot reduce recoverable information about a fixed inferential target, an exact chain-rule identity, while practical value can fall. The fourth node is the report itself. A spatial Bayesian model factorizes mislocalization into referral blur, an anatomical offset and a precision-weighted cognitive override, and reproduces phantom-limb, mirror-box and central-post-stroke reports as regimes of one equation. It is estimable under conditions the paper states as necessary or sufficient rather than assuming them; Sec. 8 gives the design that meets them. One control is reported against the paper's own interest: the transport statistic first used to measure migration scores a stationary, spreading profile as though it had travelled.
CommentsThis paper was previously the mathematical appendix of arXiv:2607.26297 and is developed here as a standalone treatment, with an introduction, a discussion of what would falsify each result, and its own glossary. arXiv:2607.26297v2 has been revised to remove that material; the two papers are companions and neither supersedes the other. 57 pages, 15 figures
