  The new discoveries of circumbinary planetary systems shed light on the
understanding of planetary system formation. Learning the architectural
properties of these systems is essential for constraining the different
formation mechanisms. We first revisit the stability limit of circumbinary
planets. Next, we focus on eclipsing stellar binaries and obtain an analytical
expression for the transit probability in a realistic setting, where finite
observation period and planetary orbital precession are included. Then,
understanding of the architectural properties of the currently observed
transiting systems is refined, based on Bayesian analysis and a series of
hypothesis tests. We find 1) it is not a selection bias that the innermost
planets reside near the stability limit for eight of the nine observed systems,
and this is consistent with a log uniform distribution of the planetary
semi-major axis; 2) it is not a selection bias that the planetary and stellar
orbits are nearly coplanar ($\lesssim 3^\circ$), and this together with
previous studies may imply an occurrence rate of circumbinary planets similar
to that of single star systems; 3) the dominance of observed circumbinary
systems with only one transiting planet may be caused by selection effects; 4)
formation mechanisms involving Lidov-Kozai oscillations, which may produce
misalignment and large separation between planet and stellar binaries, are
consistent with the lack of transiting circumbinary planets around short-period
stellar binaries, in agreement with previous studies. As a consequence of 4),
eclipse timing variations may better suit the detection of planets in such
configurations.
