  We propose and test improvements to state-of-the-art techniques of Bayeasian
statistical inference based on pseudolikelihood maximization with $\ell_1$
regularization and with decimation. In particular, we present a method to
determine the best value of the regularizer parameter starting from a
hypothesis testing technique. Concerning the decimation, we also analyze the
worst case scenario in which there is no sharp peak in the
tilded-pseudolikelihood function, firstly defined as a criterion to stop the
decimation. Techniques are applied to noisy systems with non-linear dynamics,
mapped onto multi-variable interacting Hamiltonian effective models for waves
and phasors. Results are analyzed varying the number of available samples and
the externally tunable temperature-like parameter mimicing real data noise.
Eventually the behavior of inference procedures described are tested against a
wrong hypothesis: non-linearly generated data are analyzed with a pairwise
interacting hypothesis. Our analysis shows that, looking at the behavior of the
inverse graphical problem as data size increases, the methods exposed allow to
rule out a wrong hypothesis.
