  We present a resummed mean-field approximation for inferring the parameters
of an Ising or a Potts model from empirical, noisy, one- and two-point
correlation functions. Based on a resummation of a class of diagrams of the
small correlation expansion of the log-likelihood, the method outperforms
standard mean-field inference methods, even when they are regularized. The
inference is stable with respect to sampling noise, contrarily to previous
works based either on the small correlation expansion, on the Bethe free
energy, or on the mean-field and Gaussian models. Because it is mostly
analytic, its complexity is still very low, requiring an iterative algorithm to
solve for $N$ auxiliary variables, that resorts only to matrix inversions and
multiplications. We test our algorithm on the Sherrington-Kirkpatrick model
submitted to a random external field and large random couplings, and
demonstrate that even without regularization, the inference is stable across
the whole phase diagram. In addition, the calculation leads to a consistent
estimation of the entropy of the data and allows us to sample form the inferred
distribution to obtain artificial data that are consistent with the empirical
distribution.
