  Universal scaling of entanglement estimators of critical quantum systems has
drawn a lot of attention in the past. Recent studies indicate that similar
universal properties can be found for bipartite information estimators of
classical systems near phase transitions, opening a new direction in the study
of critical phenomena. We explore this subject by studying the information
estimators of classical spin chains with general mean-field interactions. In
our explicit analysis of two different bipartite information estimators in the
canonical ensemble we find that, away from criticality both the estimators
remain finite in the thermodynamic limit. On the other hand, along the critical
line there is a logarithmic divergence with increasing system-size. The
coefficient of the logarithm is fully determined by the mean-field interaction
and it is the same for the class of models we consider. The scaling function,
however, depends on the details of each model. In addition, we study the
information estimators in the micro-canonical ensemble, where they are shown to
exhibit a different universal behavior. We verify our results using numerical
calculations of two specific cases of the general Hamiltonian.
