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SUMMARY Inverse problems play a central role in data analysis across the fields of science. Many techniques and algorithms provide parameter estimation including the best-fitting model and the parameters statistics. Here, we concern ourselves with the robustness of parameter estimation under constraints, with the focus on assimilation of noisy data with potential outliers, a situation all too familiar in Earth science, particularly in analysis of remote-sensing data. We assume a linear, or linearized, forward model relating the model parameters to multiple data sets with a priori unknown uncertainties that are left to be characterized. This is relevant for global navigation satellite system and synthetic aperture radar data that involve intricate processing for which uncertainty estimation is not available. The model is constrained by additional equalities and inequalities resulting from the physics of the problem, but the weights of equalities are unknown. We formulate the problem from a Bayesian perspective with non-informative priors. The posterior distribution of the model parameters, weights and outliers conditioned on the observations are then inferred via Gibbs sampling. We demonstrate the practical utility of the method based on a set of challenging inverse problems with both synthetic and real space-geodetic data associated with earthquakes and nuclear explosions. We provide the associated computer codes and expect the approach to be of practical interest for a wide range of applications.