We show that when a Brownian bridge is physically constrained to satisfy a canonical condition, its time evolution exactly coincides with an m-geodesic on the statistical manifold of Gaussian distributions. This identification provides a direct physical realization of a geometric concept in information geometry. It implies that purely random processes evolve along informationally straight trajectories, analogous to geodesics in general relativity. Our findings suggest that the asymmetry of informational ``distance" (divergence) plays a fundamental physical role, offering a concrete step toward an equivalence principle for information.