  The four-planet system around GJ 581 has received attention because it has
been claimed that there are possibly two additional low-mass companions as well
- one of them being a planet in the middle of the stellar habitable zone. We
re-analyse the available HARPS and HIRES Doppler data in an attempt to
determine the false positive rate of our Bayesian data analysis techniques and
to count the number of Keplerian signals in the GJ 581 data. We apply the
common Lomb-Scargle periodograms and posterior sampling techniques in the
Bayesian framework to estimate the number of signals in the radial velocities.
We also analyse the HARPS velocities sequentially after each full observing
period to compare the sensitivities and false positive rates of the two signal
detection techniques. By relaxing the assumption that the radial velocity noise
is white, we also demonstrate the consequences that noise correlations have on
the obtained results and the significances of the signals. According to our
analyses, the number of Keplerian signals favoured by the publicly available
HARPS and HIRES radial velocity data of GJ 581 is four. This result relies on
the sensitivity of the Bayesian statistical analysis techniques but also
depends on the assumed noise model. We also show that the radial velocity noise
is actually not white and that this feature has to be accounted for when
analysing radial velocities in a search for low-amplitude signals corresponding
to low-mass planets. ...
