  Consider the set of all sequences of $n$ outcomes, each taking one of $m$
values, that satisfy a number of linear constraints. If $m$ is fixed while $n$
increases, most sequences that satisfy the constraints result in frequency
vectors whose entropy approaches that of the maximum entropy vector satisfying
the constraints. This well-known "entropy concentration" phenomenon underlies
the maximum entropy method.
  Existing proofs of the concentration phenomenon are based on limits or
asymptotics and unrealistically assume that constraints hold precisely,
supporting maximum entropy inference more in principle than in practice. We
present, for the first time, non-asymptotic, explicit lower bounds on $n$ for a
number of variants of the concentration result to hold to any prescribed
accuracies, with the constraints holding up to any specified tolerance, taking
into account the fact that allocations of discrete units can satisfy
constraints only approximately. Again unlike earlier results, we measure
concentration not by deviation from the maximum entropy value, but by the
$\ell_1$ and $\ell_2$ distances from the maximum entropy-achieving frequency
vector. One of our results holds independently of the alphabet size $m$ and is
based on a novel proof technique using the multi-dimensional Berry-Esseen
theorem. We illustrate and compare our results using various detailed examples.
