  We propose a new method of solving a class of mean-field (MF) models, which
is based on the Maximum Entropy (MaxEnt) principle with additional constraints
included. Next, we show equivalence of our method when applied to the
Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism
(on the example of the single-band Hubbard model). This equivalence provides
thus an alternative justification of the results obtained within the SB
approach which, however, contains ad hoc assumptions to position it in
agreement with GA. Our approach implies that all predictions of the MF SB
method can be obtained in a simpler, transparent, and controllable manner
within GA when supplemented with the statistical-consistency conditions. We
call the method as the Statistically-consistent Gutzwiller Approximation (SGA).
Explicitly, the present formulation does not require introducing the condensed
amplitudes of auxiliary Bose fields, which do not have a direct physical
meaning and do not appear in the present formulation. Although the results of
SGA are in the present case equivalent to SB, one can improve them further by
utilizing more advanced schemes of calculating averages beyond the standard GA.
To illustrate our approach, as well as to outline its advantages over
alternative treatments of GA, we select the case of almost localized Fermi
liquid (ALFL) in two dimensions and analyze it in detail within the
tight-binding approximation. We also comment on significance of our method for
describing correlated fermions. Namely, the reasoning used here can be applied
to the corresponding MF treatment of the multiband Hubbard, the periodic
Anderson, the $t-J$, and the $t-J-U$ models. In this manner our method may be
applied for strongly correlated electron systems, optical lattices, and other
related situations.
