Data was generated from a third order model, as explained in the section
“Generating synthetic data” (Methods), and fit to pairwise maximum entropy models
and independent models. All data points correspond to averages over
marginalizations of the true distribution (see text for details). The
red points were computed directly using Eqs. (1), (3) and (4); the blue
points are the zeroth order estimates, , , and , in rows 1, 2 and 3, respectively. The three columns
correspond to , 0.029, and 0.037, from left to right. (A, B, C) (). Predictions from the perturbative expansion are in
good agreement with the measurements up to , indicating that the data is in the perturbative
regime. (D, E, F) (). Predictions from the perturbative expansion are in
good agreement with the measurements up to , indicating that the data is only partially in the
perturbative regime. (G, H, I) (). Predictions from the perturbative expansion are not
in good agreement with the measurements, even for small , indicating that the data is outside the perturbative
regime.