We present a comprehensive analysis of 10 years of HARPS radial velocities of the K2V dwarf star HD 13808, which has previously been reported to host two unconfirmed planet candidates. We use the state-of-the-art nested sampling algorithm PolyChord to compare a wide variety of stellar activity models, including simple models exploiting linear correlations between RVs and stellar activity indicators, harmonic models for the activity signals, and a more sophisticated Gaussian process regression model. We show that the use of overly-simplistic stellar activity models that are not well-motivated physically can lead to spurious `detections of planetary signals that are almost certainly not real. We also reveal some difficulties inherent in parameter and model inference in cases where multiple planetary signals may be present. Our study thus underlines the importance both of exploring a variety of competing models and of understanding the limitations and precision settings of ones sampling algorithm. We also show that at least in the case of HD 13808, we always arrive at consistent conclusions about two particular signals present in the RV, regardless of the stellar activity model we adopt; these two signals correspond to the previously-reported though unconfirmed planet candidate signals. Given the robustness and precision with which we can characterize these two signals, we deem them secure planet detections. In particular, we find two planets orbiting HD 13808 at distances of 0.11, 0.26 AU with periods of 14.2, 53.8 d, and minimum masses of 11, 10 Earth masses.
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