  Unlike the classical linear model, nonlinear generative models have been
addressed sparsely in the literature of statistical learning. This work aims to
bringing attention to these models and their secrecy potential. To this end, we
invoke the replica method to derive the asymptotic normalized cross entropy in
an inverse probability problem whose generative model is described by a
Gaussian random field with a generic covariance function. Our derivations
further demonstrate the asymptotic statistical decoupling of the Bayesian
estimator and specify the decoupled setting for a given nonlinear model.
  The replica solution depicts that strictly nonlinear models establish an
all-or-nothing phase transition: There exists a critical load at which the
optimal Bayesian inference changes from perfect to an uncorrelated learning.
Based on this finding, we design a new secure coding scheme which achieves the
secrecy capacity of the wiretap channel. This interesting result implies that
strictly nonlinear generative models are perfectly secured without any secure
coding. We justify this latter statement through the analysis of an
illustrative model for perfectly secure and reliable inference.
