  I consider the problem of deriving couplings of a statistical model from
measured correlations, a task which generalizes the well-known inverse Ising
problem. After reminding that such problem can be mapped on the one of
expressing the entropy of a system as a function of its corresponding
observables, I show the conditions under which this can be done without
resorting to iterative algorithms. I find that inverse problems are local (the
inverse Fisher information is sparse) whenever the corresponding models have a
factorized form, and the entropy can be split in a sum of small cluster
contributions. I illustrate these ideas through two examples (the Ising model
on a tree and the one-dimensional periodic chain with arbitrary order
interaction) and support the results with numerical simulations. The extension
of these methods to more general scenarios is finally discussed.
