  We discuss a characterization of complexity based on successive
approximations of the probability density describing a system by means of
maximum entropy methods, thereby quantifying the respective role played by
different orders of interaction. This characterization is applied on simple
cellular automata in order to put it in perspective with the usual notion of
complexity for such systems based on Wolfram classes. The overlap is shown to
be good, but not perfect. This suggests that complexity in the sense of Wolfram
emerges as an intermediate regime of maximum entropy-based complexity, but also
gives insights regarding the role of initial conditions in complexity-related
issues.
