This paper develops a self-parametrized Bayesian inversion to infer the spatio-temporal evolution of tsunami sources (initial sea state) due to megathrust earthquakes. To date, tsunami-source uncertainties are poorly understood, and the effect of choices such as discretization have not been studied. The approach developed here is based on a trans-dimensional self-parametrization of the sea surface, avoids regularization constraints and provides rigorous uncertainty estimation that accounts for model-selection ambiguity associated with the source discretization. The sea surface is parametrized using self-adapting irregular grids, which match the local resolving power of the data and provide parsimonious solutions for complex source characteristics. Source causality is ensured by including rupture-velocity and obtaining delay times from the Eikonal equation. The data are recorded on ocean-bottom pressure and coastal wave gauges and predictions are based on Green-function libraries computed from ocean-basin scale tsunami models for cases that include/exclude dispersion effects. The inversion is applied to tsunami waveforms from the great 2011 Tohoku-Oki (Japan) earthquake. The tsunami source is strongest near the Japan trench with posterior mean amplitudes of ~5 m. In addition, the data appear sensitive to rupture velocity, which is part of our kinematic source model.