We demonstrate that when a Brownian bridge is physically constrained to be canonical, its time evolution becomes identical to an m-geodesic on the statistical manifold of Gaussian distributions. This finding provides strong evidence that, akin to general relativity where free particles follow geodesics, purely random processes also follow ``straight lines" defined by the geometry of information. This geometric principle is a direct consequence of the dually flat structure inherent to information geometry, originating from the asymmetry of informational ``distance" (divergence) leading to the violation of metric compatibility. Our results suggest a geometric foundation for randomness and open the door to an equivalence principle for information.

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