We are motivated by applications that need rich model classes to represent
them. Examples of rich model classes include distributions over large,
countably infinite supports, slow mixing Markov processes, etc. But such rich
classes may be too complex to admit estimators that converge to the truth with
convergence rates that can be uniformly bounded over the entire model class as
the sample size increases (uniform consistency). However, these rich classes
may still allow for estimators with pointwise guarantees whose performance can
be bounded in a model dependent way. The pointwise angle of course has the
drawback that the estimator performance is a function of the very unknown model
that is being estimated, and is therefore unknown. Therefore, even if the
estimator is consistent, how well it is doing may not be clear no matter what
the sample size is. Departing from the dichotomy of uniform and pointwise
consistency, a new analysis framework is explored by characterizing rich model
classes that may only admit pointwise guarantees, yet all the information about
the model needed to guage estimator accuracy can be inferred from the sample at
hand. To retain focus, we analyze the universal compression problem in this
data driven pointwise consistency framework.Comment: Working paper. Please email authors for the current versio