  It is a challenge to manage infinite- or high-dimensional data in situations
where storage, transmission, or computation resources are constrained. In the
simplest scenario when the data consists of a noisy infinite-dimensional
signal, we introduce the notion of local \emph{effective dimension} (i.e.,
pertinent to the underlying signal), formulate and study the problem of its
recovery on the basis of noisy data. This problem can be associated to the
problems of adaptive quantization, (lossy) data compression, oracle signal
estimation. We apply a Bayesian approach and study frequentists properties of
the resulting posterior, a purely frequentist version of the results is also
proposed. We derive certain upper and lower bounds results about identifying
the local effective dimension which show that only the so called
\emph{one-sided inference} on the local effective dimension can be ensured
whereas the \emph{two-sided inference}, on the other hand, is in general
impossible. We establish the \emph{minimal} conditions under which two-sided
inference can be made. Finally, connection to the problem of smoothness
estimation for some traditional smoothness scales (Sobolev scales) is
considered.
