Assessing complexity by means of maximum entropy models

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.