  Context. Serendipitous X-ray surveys have proven to be an efficient way to
find rare objects, for example tidal disruption events, changing-look active
galactic nuclei (AGN), binary quasars, ultraluminous X-ray sources (ULXs), and
intermediate mass black holes. With the advent of very large X-ray surveys, an
automated classification of X-ray sources becomes increasingly valuable.Aims.
This work proposes a revisited naive Bayes classification of the X-ray sources
in the Swift-XRT and XMM-Newton catalogs into four classes -- AGN, stars, X-ray
binaries (XRBs) and cataclysmic variables (CVs) -- based on their spatial,
spectral and timing properties and their multiwavelength counterparts. An
outlier measure is used to identify objects of other natures. The classifier is
optimized to maximize the classification performance of a chosen class (here
XRBs) and it is adapted to data mining purposes.Methods. We augmented the X-ray
catalogs with multiwavelength data, source class, and variability properties.
We then built a reference sample of about 25000 X-ray sources of known nature.
From this sample the distribution of each property was carefully estimated and
taken as reference to assign probabilities of belonging to each class. The
classification was then performed on the whole catalog, combining the
information from each property.Results. Using the algorithm on the Swift
reference sample we retrieved 99%, 98%, 92% and 34% of AGN, stars, XRBs, and
CVs, respectively, and the false positive rates are 3%, 1%, 9% and 15%. Similar
results are obtained on XMM sources. When applied to a carefully selected test
sample, representing 55% of the X-ray catalog, the classification gives
consistent results in terms of distributions of source properties. A
substantial fraction of sources not belonging to any class is efficiently
retrieved using the outlier measure, as well as AGN and stars with properties
deviating from the bulk of their class. Our algorithm is then compared to a
random forest method; the two showed similar performances but the algorithm
presented in this paper improved insight into the grounds of each
classification.Conclusions. This robust classification method can be tailored
to include additional or different source classes and can be applied to other
X-ray catalogs. The transparency of the classification compared to other
methods makes it a useful tool in the search for homogeneous populations or
rare source types, including multi-messenger events. Such a tool will be
increasingly valuable with the development of surveys of unprecedented size,
such as LSST, SKA and Athena, and the search for counterparts of
multi-messenger events.
