We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are shown to arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework, the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance.