Some problems associated with the use of the maximum entropy principle, namely, (i) possible divergence of the series that is exponentiated, (ii) input-dependent asymptotic behaviour of the density function resulting from the truncation of the said series, and (iii) non-vanishing of the density function at the boundaries of a finite domain are pointed out. Prescriptions for remedying the aforesaid problems are put forward. Pilot calculations involving the ground quantum eigenenergy states of the quartic oscillator, the particle-in-a-box model, and the classical Maxwellian speed and energy distributions lend credence to our approach.