  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 one's 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.
