<p>Angle-resolved photoemission spectroscopy is a powerful experimental technique for studying anisotropic many-body interactions through the electron spectral function. Existing attempts to decompose the spectral function into non-interacting dispersions and electron-phonon, electron-electron, and electron-impurity self-energies rely on linearization of the bands and manual assignment of self-energy magnitudes. Here, we show how self-energies can be extracted consistently for curved dispersions. We extend the maximum-entropy method to Eliashberg-function extraction with Bayesian inference, optimizing the parameters describing the dispersions and the magnitudes of electron-electron and electron-impurity interactions. We compare these novel methodologies with state-of-the-art approaches on model data, then demonstrate their applicability with two high-quality experimental data sets. With the first set, we identify the phonon modes of a two-dimensional electron liquid on TiO₂-terminated SrTiO₃. With the second set, we obtain unprecedented agreement between two Eliashberg functions of Li-doped graphene extracted from separate dispersions. We release these functionalities in the novel Python code <span>x</span>ARPES.</p>