Current three-dimensional, data-based models for the terrestrial exosphere have been derived from measurements of optically thin Lyman-alpha (Ly-α) emissions scattered by neutral hydrogen atoms. Such models are only valid for the middle exospheric region (3-8 Earth radii geocentric distances) since the orbital paths of the space-based platforms used to acquire Ly-α radiance were located within the exosphere, thus precluding the proper detection of the faint outer exospheric emission. Notwithstanding, accurate specifications of density distributions beyond 8 RE are needed to support comprehensive studies of the solar-terrestrial interactions. Two upcoming missions, the Solar wind Magnetosphere-Ionosphere Link Explorer (SMILE) and the Lunar Environment Heliospheric X-ray Imager (LEXI), will image the Earth’s magnetosheath in soft X-rays, and neutral densities are crucial to extract ion distributions through inversion of the acquired images. This work develops a technique to estimate the Earth’s outer exospheric density distributions using far-ultraviolet wide-field data acquired by the Lyman-Alpha Imaging Camera (LAICA) onboard the Proximate Object Close Flyby with Optical Navigation mission. Our approach formulates an inverse problem based on the linearity between measurements of scattered Ly-α flux and the local atomic hydrogen density, which is solved using the Bayesian approach known as Maximum a posteriori estimation. We use the LAICA image to derive global, 3-D hydrogen density distributions at 6-35 RE geocentric distances. We find that the spatial structure of the outer exosphere agrees well with the predictions of radiation pressure theory. Further, we find that the mean hydrogen density at 10 RE subsolar point is 26.51 atoms/cm³.