  Robust position reconstruction is paramount for enabling discoveries in
astroparticle physics as backgrounds are significantly reduced by only
considering interactions within the fiducial volume. In this work, we present
for the first time a method for position reconstruction using a Bayesian
network which provides per interaction uncertainties. We demonstrate the
utility of this method with simulated data based on the XENONnT detector
design, a dual-phase xenon time-projection chamber, as a proof-of-concept. The
network structure includes variables representing the 2D position of the
interaction within the detector, the number of electrons entering the gaseous
phase, and the hits measured by each sensor in the top array of the detector.
The precision of the position reconstruction (difference between the true and
expectation value of position) is comparable to the state-of-the-art methods --
an RMS of 0.69 cm, ~0.09 of the sensor spacing, for the inner part of the
detector (<60 cm) and 0.98 cm, ~0.12 of the sensor spacing, near the wall of
the detector (>60 cm). More importantly, the uncertainty of each interaction
position was directly computed, which is not possible with other reconstruction
methods. The method found a median 3-$\sigma$ confidence region of 11 cm$^2$
for the inner part of the detector and 21 cm$^2$ near the wall of the detector.
We found the Bayesian network framework to be well suited to the problem of
position reconstruction. The performance of this proof-of-concept, even with
several simplifying assumptions, shows that this is a promising method for
providing per interaction uncertainty, which can be extended to energy
reconstruction and signal classification.
