The probability distribution function for an out of equilibrium system may sometimes be approximated by a physically motivated "trial" distribution. A particularly interesting case is when a driven system (e.g., active matter) is approximated by a thermodynamic one. We show here that every set of trial distributions yields an inequality playing the role of a generalization of the second law. This suggests a variational procedure that may be implemented numerically and even experimentally to approximate the distribution function in terms of effective interactions. The fluctuation relation behind this inequality, -a natural and practical extension of the Hatano-Sasa theorem-, does not rely on the a priori knowledge of the stationary probability distribution.