Transitional inference is an empiricism concept, rooted and practiced in clinical medicine since ancient Greek. Knowledge and experiences gained from treating one entity are applied to treat a related but distinctively different one. This notion of "transition to the similar" renders individualized treatments an operational meaning, yet its theoretical foundation defies the familiar inductive inference framework. The uniqueness of entities is the result of potentially an infinite number of attributes (hence $p=\infty$), which entails zero direct training sample size (i.e., $n=0$) because genuine guinea pigs do not exist. However, the literature on wavelets and on sieve methods suggests a principled approximation theory for transitional inference via a multi-resolution (MR) perspective, where we use the resolution level to index the degree of approximation to ultimate individuality. MR inference seeks a primary resolution indexing an indirect training sample, which provides enough matched attributes to increase the relevance of the results to the target individuals and yet still accumulate sufficient indirect sample sizes for robust estimation. Theoretically, MR inference relies on an infinite-term ANOVA-type decomposition, providing an alternative way to model sparsity via the decay rate of the resolution bias as a function of the primary resolution level. Unexpectedly, this decomposition reveals a world without variance when the outcome is a deterministic function of potentially infinitely many predictors. In this deterministic world, the optimal resolution prefers over-fitting in the traditional sense when the resolution bias decays sufficiently rapidly. Furthermore, there can be many "descents" in the prediction error curve, when the contributions of predictors are inhomogeneous and the ordering of their importance does not align with the order of their inclusion in prediction.

中文翻译：

---

用于逼近无限 p 零 n 的多分辨率理论：过渡推理、个性化预测和没有偏差方差权衡的世界

过渡推理是一个经验主义概念，自古希腊以来就在临床医学中扎根和实践。从处理一个实体中获得的知识和经验可用于处理相关但截然不同的实体。这种“过渡到相似”的概念使个体化治疗具有操作意义，但其理论基础与熟悉的归纳推理框架背道而驰。实体的唯一性是潜在无限数量属性的结果（因此 $p=\infty$），这需要零直接训练样本大小（即 $n=0$），因为真正的豚鼠不存在。然而，关于小波和筛法的文献通过多分辨率 (MR) 的角度提出了一种有原则的近似理论，用于过渡推理，我们使用分辨率级别来衡量对终极个性的近似程度。MR 推理寻求对间接训练样本进行索引的主要分辨率，它提供足够的匹配属性以增加结果与目标个体的相关性，同时仍然积累足够的间接样本大小以进行稳健的估计。从理论上讲，MR 推理依赖于无限项 ANOVA 类型的分解，通过分辨率偏差的衰减率作为主要分辨率级别的函数，提供了一种对稀疏性进行建模的替代方法。出乎意料的是，当结果是潜在无限多个预测变量的确定性函数时，这种分解揭示了一个没有方差的世界。在这个确定性的世界里，当分辨率偏差衰减得足够快时，最佳分辨率更喜欢传统意义上的过度拟合。此外，当预测变量的贡献不均匀且其重要性的顺序与其包含在预测中的顺序不一致时，预测误差曲线中可能存在许多“下降”。