  Transitional inference is an empiricism concept, rooted and practiced in
clinical medicine since ancient Greece. 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.
