We expand a method for seismic moment tensor inversion using probabilistic Bayesian inference, which yields parameter uncertainties and includes a thorough treatment of noise in the data, to include additional noise parameters that weight the contributions of particular stations. In a synthetic test, we show that having individual noise parameters for each station gives an optimal fit to the data. The noise determines the level of data fit at each station and in turn weights their contribution in the final solution. Apart from the noise level, an empirically determined data covariance matrix accounts for noise correlations present in waveform data. This improves the estimate of the centroid location and the non‐double‐couple (non‐DC) components. We apply the method to two earthquakes, one from a volcanic (Long Valley caldera [LVC]) and another from a geothermal (The Geysers) environment in California, which are likely to have non‐DC components in the source mechanism. We confirm a significant isotropic (ISO) component for the LVC earthquake. Implementing a cosine data covariance matrix reduces the trade‐off between the ISO and compensated linear vector dipole components for The Geysers earthquake and yields considerably higher non‐DC components. This shows the importance of adequate noise treatment for earthquakes in complex tectonic environments.