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Transitional inference is an empiricism concept, rooted and practiced inclinical medicine since ancient Greece. Knowledge and experiences gained fromtreating one entity are applied to treat a related but distinctively differentone. This notion of "transition to the similar" renders individualizedtreatments an operational meaning, yet its theoretical foundation defies thefamiliar inductive inference framework. The uniqueness of entities is theresult of potentially an infinite number of attributes (hence $p=\infty$),which entails zero direct training sample size (i.e., $n=0$) because genuineguinea pigs do not exist. However, the literature on wavelets and on sievemethods suggests a principled approximation theory for transitional inferencevia a multi-resolution (MR) perspective, where we use the resolution level toindex the degree of approximation to ultimate individuality. MR inference seeksa primary resolution indexing an indirect training sample, which providesenough matched attributes to increase the relevance of the results to thetarget individuals and yet still accumulate sufficient indirect sample sizesfor robust estimation. Theoretically, MR inference relies on an infinite-termANOVA-type decomposition, providing an alternative way to model sparsity viathe decay rate of the resolution bias as a function of the primary resolutionlevel. Unexpectedly, this decomposition reveals a world without variance whenthe outcome is a deterministic function of potentially infinitely manypredictors. In this deterministic world, the optimal resolution prefersover-fitting in the traditional sense when the resolution bias decayssufficiently rapidly. Furthermore, there can be many "descents" in theprediction error curve, when the contributions of predictors are inhomogeneousand the ordering of their importance does not align with the order of theirinclusion in prediction.

A Multi-resolution Theory for Approximating Infinite-$p$-Zero-$n$:  Transitional Inference, Individualized Predictions, and a World Without  Bias-Variance Trade-off