(abridged) We develop Bayesian methods and detection criteria for orbital fitting, and revise the detectability of exoplanets in light of the in-flight properties of Gaia. Limiting ourselves to one-planet systems as a first step of the development, we simulate Gaia data for exoplanet systems over a grid of S/N, orbital period, and eccentricity. The simulations are then fit using Markov chain Monte Carlo methods. We investigate the detection rate according to three information criteria and the delta chi^2. For the delta chi^2, the effective number of degrees of freedom depends on the mission length. We find that the choice of the Markov chain starting point can affect the quality of the results; we therefore consider two limit possibilities: an ideal case, and a very simple method that finds the starting point assuming circular orbits. Using Jeffreys scale of evidence, the fraction of false positives passing a strong evidence criterion is < ~0.2% (0.6%) when considering a 5 yr (10 yr) mission and using the Akaike information criterion or the Watanabe-Akaike information criterion, and <0.02% (<0.06%) when using the Bayesian information criterion. We find that there is a 50% chance of detecting a planet with a minimum S/N=2.3 (1.7). This sets the maximum distance to which a planet is detectable to ~70 pc and ~3.5 pc for a Jupiter-mass and Neptune-mass planet, respectively, assuming a 10 yr mission, a 4 au semi-major axis, and a 1 M_sun star. The period is the orbital parameter that can be determined with the best accuracy, with a median relative difference between input and output periods of 4.2% (2.9%) assuming a 5 yr (10 yr) mission. The median accuracy of the semi-major axis of the orbit can be recovered with a median relative error of 7% (6%). The eccentricity can also be recovered with a median absolute accuracy of 0.07 (0.06).
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